What are some other ways in which a parabola is "between an ellipse and a hyperbola"? What are some other ways in which a parabola is "between an ellipse and a hyperbola"?
On page 122 of Gilbert Strangs calculus text he writes:
"Throughout mathematics, parabolas are on the border between ellipses and hyperbolas."
Here are three ways in which we can think of a parabola as being inbetween an ellipse and a hyperbola:

*

*If we cut a cone with a horizontal plane, we get a circle. When we tilt the plane slightly, we get an ellipse. If we tilt the plane a lot, we get a hyperbola. When we tilt the plane so that its angle matches the slope of the cone, we get a parabola.


*The equation $Ax^2+Bxy+Cy^2=1$ produces a hyperbola if $B^2 > 4ac$, an ellipse if $B^2-4AC <0$ and a parabola if $B^2-4ac=0$


*In the polar form $r=\frac\ell{1+e\cos\theta}$, we pass smoothly through a parabola when the eccentricity $e$ passes through $1$ with a fixed semi latus rectum $\ell$.
I'm suspecting there is at least one more way to understand why we think of a parabola as being inbetween an ellipse and a hyperbola, perhaps in terms of foci. Strang writes that the "second foci of a parabola" is located at ininfinity, and I'm not quite sure why this type of thinking makes sense and if we can somehow relate this foci at infinity to being an inbetween case of defining ellipses and hyperbola by their foci.
 A: Another case which comes to mind goes back to Apollonius of Perga, and the reason he gave the conics those three names we still use nowadays.
If we choose as $x$ axis the major axis of the conic and as $y$ axis the line tangent to the conic at one of its intersections with $x$ axis, the equations of parabola, ellipse and hyperbola can be respectively written as:
$$
y^2=px,\quad
y^2=px -{p\over d}x^2,\quad
y^2=px +{p\over d}x^2,
$$
where $p$ is the latus rectum of the conic and $d$ the length of major axis. That is (quoting from Heath's comment to Conics):

if a perpendicular be erected to the diameter at that extremity of it
from which $x$ is measured and of length $p$, then $y^2$ is equal
[parabole] to a rectangle of breadth $x$ and "applied" to the
perpendicular of length p, or falling short [ellipsis] or exceeding
[hyperbole] by a rectangle similar and similarly situated to that
contained by $p$ and $d$.

In the figure below one can see that in the case of an ellipse: the area of the red square ($y^2$) is equal to the area of the blue rectangle, obtained by subtracting from rectangle $px$ the green rectangle, which is "similar and similarly situated" to the big rectangle $pd$.

A: In projective geometry, hyperbolas, ellipses, and parabola are essentially the same thing, are can be transformed into each other with a change of coordinates.  A circle that is inside the line at infinity looks like a regular circle. One that intersects the line at infinity looks like a hyperbola. If it's tangent to the line at infinity, it looks like a parabola.
A: Don't know if this is really an answer, but when in doubts...
4
In a parabola the two arms "become" (read: approach) parallel to each other whereas in a hyperbola they do not.
5
All parabolas are of the same shape no matter what the size; all hyperbolas are of different shapes.
6
When a set of points present in a plane are equidistant from the directrix, a given straight line, and are equidistant from the focus, a given point which is fixed, it is called a parabola.
When the difference of distances between a set of points present in a plane to two fixed foci or points is a positive constant, it is called a hyperbola. (This is really textbook, not sure it helps here).
I suspect that maybe only point 4 could result helpful in seeing why parabolas are in between (considering that ellipses degenerate on a single straight line). Perhaps not. In case, I will delete.
A: If you've ever used Kerbal Space Program or other spaceflight simulator (or work for NASA as an actual rocket scientist), you'll notice that your rocket's velocity around a planet will determine the shape of its projected orbit.  Sometimes you'll get an elliptical orbit.  Sometimes you'll get a hyperbolic "orbit".  If you arrange things just right, you can get a parabolic orbit, and that speed (known as escape velocity) will be exactly at the transition point between elliptical-orbit and hyperbolic-orbit speeds.
A: There is one fact which explains all these answers (and more): Conics (up to translation, rotation, and dilation) are defined by a single parameter: eccentricity (known as $e$).  In any definition of conics, $e$ must show up, though in some forms its easier to see than others. When $e < 1$, we have an ellipse; $e > 1$, a hyperbola, and for $e$ exactly $1$, a parabola.  It is this fact that underlies the other answers - and much more:

*

*In polar coordinates, $e$ is easy to see: $r = \frac 1 {1 + e \cos \theta}$ [Point #3 from OP]

*Conics are the intersection of a cone $z = \pm \sqrt {x^2 + y^2}$ by the plane $z = ex + b$ [Point #1 from OP, with $e$ identified]

*Conics are the locus of points whose ratio of the distance from a given point (focus) and from a given line (directrix) is $e$ [not mentioned by others]

*Eccentricity is determined by the ratio of velocity to escape velocity: $e = 2 (\frac v {v_e})^2 - 1$ [Dan's answer]

*Since a conic's shape is solely determined by $e$, all parabolas must have the same shape (up to dilation) [from EnricoM's answer]

*$e$ is a function of $A, B, C$ in $Ax^2 + Bxy + Cx = 1$ though the relationship seems messy [Point #2 from OP]

*In Intelligenti pauca's answer, I believe $e = \sqrt{1 + \frac p d}$
In fact, all properties of a non-degenerate conic are solely determined by $e$ (up to dilation, rotation, and translation) -- they simply have no other parameters! Conics behave one way - in properties or in the means of generating them - when $e < 1$, a different way when $e > 1$, and $e = 1$ is in between.
Extending this further, and casting rigor to the wind, as $e \to \infty$, we get (pairs of) lines.  So $r = \frac 1 {\cos \theta}$ is a line in polar coordinates; points on the directrix line itself admit ratio $e = \frac d 0 \to \infty$; and as $v \to \infty$, the trajectory approaches a straight line, completely bypassing the planet.
