Are functions with odd-numbered exponents parabolas? Or are only functions with even-numbered exponents called 'parabolas' and, if so, is there another name for functions with odd-numbered exponents like cubics and quintics?
 A: Only functions where the highest degree term has degree $2$ give rise to graphs called parabolas. The graph of $x\mapsto x^4$ is not a parabola.
I don't think there is a name for higher degree curves, other than "cubic curve", "quartic curve", "quintic curve", and so on. They just don't have the same tradition behind them, and as such they haven't received fancy names.
A: A parabola is a particular kind of curve. The standard examples are the graphs of the quadratic equations $y=ax^2 + bx + c$ in the usual coordinate system. The Greeks knew them as sections of a cone.
The graph of $y=x^4$ has a roughly similar shape, but it's not a parabola.
A: Only second degree polynomials $$y=ax^2+bx+c$$ where $a\ne 0$ are parabolas.
For third degree polynomials sometimes $S$ shaped is used but cubic is more common.
For higher degrees quartic, quintic and so forth are used.
 I call them "polymomials of n-th degree" which serves the purpose  without using fancy names. 
A: A function is a weaker definition of a parabola (as in there are parabolas that are not functions since parabolas can map the same input to two different points on the plane. A parabola has a strict definition that isn't quite as intuitive (see Wolfram link).
However, to answer your question more directly, no. Parabolas come in all different shapes and forms, but the precise definition follows from "conic forms". This is a fancy definition for just a curve you'd get if you took any subsection of a cone. Even with the strict mathematical definition of conic forms, no odd functions fall under the parabolas.
Most parabolas fall under the form of $ax^2 + bx + c = 0$, however there are rotations to this that also count as parabolas (such as the function $(y+3)^2=12(x-1)$). You can think of parabolas as curves that satisfies $ax^2 + bx + c = 0$ where $a$ is non-zero, along with any linear transformations of it (translation, rotation and reflection).
A: The graph of a function may change direction one less times than its power. For example a parabola may go down, then up, or vice versa. A cubic may go up-down-up or up-sideways-up or some other plot with two changes of directional $curve$. A quartic will change direction three times and so on. As Ethan Bolker noted, they can have a similar shape but not quite. We can see here how there are $3$ curve changes in a quartic but, stepping farther back, the whole think looks similar to a parabola.
Still, only a quadratic produces a parabola.
