Is an empty parenthesis a valid mathematical expression? Is using an empty parenthesis valid? 
For example, $15+()=15$. 
What is the meaning if it is valid? I need an academic reference to validate this.
 A: It's certainly not conventional, and it's hard to think of any occasion when one might want to use such notation explicitly. That said, it would not be illogical to define () as zero, just as the empty set is sometimes written $\{\}$. As a precedent, an empty sum, such as $\sum_{k=1}^0x_k$, is defined to be zero by a standard convention.
A: Anything is valid if you define it. Mathematicians tend to not only use different notation for the same thing, but often the same notation to denote different things. That's all right as long as they define up front what a given notation is supposed to mean. In many cases, the meaning is so commonplace that all mathematicians agree upon it. You'd find no serious debates on what $3+4$ could possibly mean.
But for other notation, things soon get far less obvious. What is $ab$? It could be the variable $a$ multiplied by the variable $b$. It could be a monomial in a multivariate polynomial ring over the indeterminates $a$ and $b$ (and perhaps others). It could be the composition of two functions, i.e. what you get if you first apply $b$ and then $a$. It could be the concatenation of words. It could be a hundred other things, and often the lines between these concepts are quite blurry, so it could be many of these things at once. So unless the intended meaning of such a notation is clear from context, the author should clarify it using a definition, lest his formulas become ambiguous.
In your case, there is no universally agreed meaning for $()$ which would make sense in a context of addition. Parentheses are used for very many things in mathematics, among them the ordering of operations Newb indicated and the collection of tuples Hagen mentioned. But for operation ordering, an empty expression inside the parentheses makes no sense, while for tuples you'd have to clarify what meaning you attach to the plus sign if the operands are as you stated them. The comment by Yves gives one possible and sensible interpretation, but taken together the interpretation is more like guessing what this could possibly mean, and notation which relies on this kind of guesswork is bad notation.
A: We usually denote $n$-tuples in the form $(a_1,\ldots,a_n)$, so for example $(x,y,z)$ is a triple, $(x,y)$ is a pair, somewhat redundantly $(x)$ could be called a one-tuple and $()$ a (in fact: the) zero-tuple. 
A: In mathematical logic, and computer programming languages like Haskell, F#, OCaml, and Scala, () represents the (single possible) value of the Unit type, which is used to denote a function that returns no value.
It is not valid to add a number with this representation of ().
A: Generally, no.
But you could say:

Let us denote something with ().

... and start using (), if you think this would convey your point.
There is no need for academic reference validating this, it is the matter of basic author freedom.
Let's say you want to wear a violet tie with blue dots. You don't ask if there is a law permitting you to do this.
However, I would advise you not to use it, it creates confusion.
A: We use parentheses to indicate the order of operations.
To refer to your example: the operator $+$ takes two arguments, in the form of $a+b$. You can think of $+$ as a function that takes two variables. 
In this example, the $+$ is missing the second argument: there's nothing there. And $15 + $ isn't a valid mathematical statement -- it's not equal to anything, because the $+$ operation requires two variables.
An empty pair of parentheses thus doesn't really mean anything -- it just contains nothing. This destroys any equation it is placed inside -- because $()$ is just nothing, not even $0$, and mathematical operations aren't defined on nothing.
A: Notation is based on common acknowledgement. Sure you can invent any random thing, for example you can even draw a monkey instead of parenthesis. But this makes people confused about the notation.
If here you mean a variable, usually people put a letter here. When the letters are not used up (or not for a specific purpose, eg. Greek letter for angle, an object usually denoted by a specific letter), put a Latin letter makes most sense. $x$ usually means an unknown.
Back to the case in your question. From my previous context, I assume you want to put a variable or an unknown there. But after I posted, I suddenly came up with an idea that what if you mean $0$ by $()$? Because why not?
