Is there a name for $f(x) = |x|^n \cdot \operatorname{sgn}(x)$?
There are many instances (eg Handbook of Computational and Numerical Methods in Finance, p. 42, Eq. 2.25) where it is simply called the "signed power"  "signed power-$p$ function" or "signed power law". 

Some instances are more common. The term "signed square-root" seems to be used, see for instance in this Simulink documentation: 


  Square root of the absolute value of the input, multiplied by the sign of the input.


with the signed square as an inverse. It is used in the realm  of statistical distributions, contrast enhancement, or on data features (like SURF) as a compander, close in spirit to the A-law or the mu-law. 

For parabolas (and even powers in general), the rewriting into $x\to x.|x|$ could be called a split parabola (see Selected solutions, HW 1, due January 22, 2009, an instance of split functions).