What is... A Parsimonious History? Interpreting historical mathematicians involves a recognition of the fact that most of them viewed the continuum as not being made out of points. Rather they viewed points as marking locations on a continuum taken more or less as a primitive notion. Modern foundational theories starting around 1870 are based on a continuum made of points and therefore cannot serve as a basis for interpreting the thinking of the earlier mathematicians as far as the foundations are concerned. 
What one can however seek to interpret are the techniques and procedures (rather than foundations) of the earlier authors, using techniques and procedures available in modern frameworks. In the case of analysis, the modern frameworks available are those developed by Weierstrass and his followers around 1870 and based on an Archimedean continuum, as well as more recently those developed by Robinson and his followers, and based on a continuum containing infinitesimals, and other frameworks such as the one developed by Lawvere, Kock, and others.
I was therefore a bit puzzled by the following comment by a historian expressed here: 
Recently there have been attempts to argue that Leibniz, Euler, and even Cauchy could have been thinking in some informal version of rigorous modern non-standard analysis, in which infinite and infinitesimal quantities do exist. However, a historical interpretation such as the one sketched above that aims to understand Leibniz on his own terms, and that confers upon him both insight and consistency, has a lot to recommend it over an interpretation that has only been possible to defend in the last few decades. It is parsimonious and requires no expert defence for which modern concepts seem essential and therefore create more problems than they solve (e.g. with infinite series). The same can be said of non-standard readings of Euler; etc. 
Question 1. Is this historian choosing one foundational framework over another in interpreting the techniques and procedures of the historical authors? 
Question 2. What exactly is a Parsimonious History?
Question 3. Gray and Bottazzini reportedly make a rather poetic proposal in the following terms: "The best policy is to read on in a spirit of dialogue with the earlier authors." The proposal of such a conversation with, say, Euler sounds intriguing. I am only wondering about Gray's comment here that "Euler’s attempts at explaining the foundations of calculus in terms of differentials, which are and are not zero, are dreadfully weak." Isn't such an opening line in a conversation likely to be a conversation-stopper?
Question 4. In connection with the work of Laugwitz mentioned by one of the responders, one could ask why Gray does not cite explicitly the work of any of the authors that he wishes explicitly to criticize for using modern infinitesimals? Laugwitz's article (Laugwitz, Detlef. Definite values of infinite sums: aspects of the foundations of infinitesimal analysis around 1820. Arch. Hist. Exact Sci.  39  (1989),  no. 3, 195–245) appeared in Archive for History of Exact Sciences, clearly a reputable journal since Jeremy Gray happens to be its Editor-in-Chief. Similarly, the article McKinzie, Mark; Tuckey, Curtis. Hidden lemmas in Euler's summation of the reciprocals of the squares. Arch. Hist. Exact Sci.  51  (1997),  no. 1, 29–57 appeared in the same journal and exploited Robinson's framework to clarify some of Euler's procedures; it, too, is being stonewalled by the Editor-in-Chief.
 A: 
Note: [2016-06-01] Please note that OP has changed the title which was:
How can a modern historian interpret historical mathematicians?
and he has added question 2. This answer addresses the original question which is at the time question 1.


To me it seems Jeremy Gray, the author of the referenced book, put some words of warning that interpreting the work of the old masters in  terms of one or two modern concepts only  is too often superficial and not appropriate to fully grasp what was really going on.
If we look at the last sentence of OPs cited paragraph we can see that the historical development is more than that.

From section 1.5: But it is to say that the foundations of the calculus were for at least two centuries the subject of shifting, partial, and largely coherent speculations that form the opening chapters of the history of analysis.


To better see the author's position I like to cite from Hidden Harmony-Geometric Fantasies: The Rise of Complex Function Theory written by Jeremy Gray and Umberto Bottazzini.

From the Introductory Section: In any history of ideas, the historian seeks to show how things once thought about in one way became thought about in another. As complex function theory developed many ideas were first introduced naively and only slowly refined.
Definitions were lacking, and when provided were sometimes inadquate by later standards. Moreover precision, when it became available, could be misleading: mathematicians on occasion offer a clear definition with very few ideas about its deepest implications - as the example of continuity in real analysis shows.
Sometimes these problems can be confronted directly, as with the very definition of an analytic function, but more often one has to ride out a long period of some vagueness.
Let us note some specific issues: Cauchy, for example, often used the phrases continuous and finite and continuous very loosely to mean something like complex analytic. Similar problems occcur with counting roots according to their multiplicities, with $\lim$ versus $\limsup$, and points of infinity and poles. ...

Later on the authors continue with their preferred approach to the history of mathematics and mathematicians:

There is therefore no truly satisfactory way to represent the originial ideas of mathematicians when they are like this. To say nothing is to produce confusion. To silently bring them into line with modern standards not only introduces anachronisms but also brings in historical falsehoods and nullifies the purpose of history.
To correct them in more than the most egregious cases is to encumbeer genuine blunders and thereby diminish the work of major mathematicians.
The best policy is to read on in a spirit of dialogue with the earlier authors, aware, as one might be, of the limitations and false implications of their papers and books, and waiting to see when, if at all in the period, better light was shone on the subject. In this way one can grapple with more of the complexity, and the drama, of the past.

A: Comment
What about Detlef Laugwitz's comments in his: Infinitely small quantities in Cauchy's textbooks, Hist.Math. 14(1987), no.3, 258–274 [that you cited elsewhere in SE]:

As a historian of mathematics one cannot but take an author’s own intentions
  and reasons seriously: Infinitely small quantities are fundamental in Cauchy’s
  analysis, they are compatible with rigor, and they produce simplicity.
[Cauchy's theorems on continuity and convergence] are incorrect when interpreted in the by now common conceptual framework of analysis (which obviously cannot have been Cauchy’s framework). Both theorems become correct as soon as one adds assumptions on uniformity (which, at least in the form by now common, were never used by Cauchy). The theorems are correct in any of the modern theories of infinitesimals (which, apart from being unknown to Cauchy, lack the “simplicity of infinitesimals,” at least in the version of Robinson).
The three attitudes mentioned (Cauchy erred; Cauchy forgot about essential
  assumptions; Cauchy was correct, but only when put against a modern background) are unsatisfactory from the point of view of a historian. [...] The only satisfactory attitude should be: Try and understand Cauchy’s theorems and their proofs from his own concepts. 


Note: according to my understanding of J.Gray's point of view, a "parsimonious history" is an approach aimed at understanding past theories and concepts "in their original environment", avoiding to overload them with recent developments.
A: This is in response to a claim by Gray and Bottazzini that allegedly "Definitions were lacking". So what? It is a basic (meta)mathematical problem: 


*

*you are given a proof that contains no definitions (or, perhaps, refers to excessively narrow definitions)

*recover the definitions.


This is what Bourbaki was doing when he was recovering real definitions from the boring and repetitive proofs of the early functional analysis. 
Why this approach cannot be used as a tool for analysis of the history of mathematics? Mathematicians of the past were using unstated definitions and unstated assumptions. Why are research mathematicians of nowadays are forbidden from doing some kind of reverse engineering and recovering their ancient  colleagues' definitions and assumptions?
www.borovik.net/selecta
A: Gray's "parsimonious" argument could be termed the Gray sword analogously with the Occam razor, and if applied in the context of a proper focus on procedures would in fact yield the opposite result of the one Gray seeks.
Consider for example Cauchy's definition of continuity, namely an infinitesimal change $\alpha$ in the variable $x$ always produces an infinitesimal change $f(x+\alpha)-f(x)$ in the function. In a modern infinitesimal framework one copies this over almost verbatim to get a precise definition of continuity. 
If one wishes to work in a traditional Weierstrassian framework, one needs to interpret Cauchy's definition as "really" saying that, for example, for every epsilon there is a delta such that for every $x$, etc.
Such logical complexity involving alternating quantifiers will surely fall by the (Gray) sword. Alternatively, one could seek to interpret Cauchy by means of sequences, which is not much better because Cauchy explicitly says in defining an infinitesimal that a sequence becomes an infinitesimal (rather than an infinitesimal being a sequence). So apparently Gray should be saying the following: 
Since Boyer (at least) there have been attempts to argue that Leibniz, Euler, and even Cauchy could have been thinking in some informal version of rigorous modern Weierstrassian analysis. However, a historical interpretation such as the one sketched above that aims to understand Leibniz on his own terms, and that confers upon him both insight and consistency, has a lot to recommend it over an interpretation that has only been possible to defend since Weierstrass came along. It is parsimonious and requires no expert defence for which modern alternating quantifiers seem essential and therefore create more problems than they solve. 
