The ReLU activation function should take care of this.

ReLU works by fitting short, straight lines to approximate curves. That should be able to create a parabola. You will have performance suffer for inputs with very large absolute values, but we know that models won't be perfect.

I was thinking that one hidden layer could take care of this, but reading about the universal approximation theorem (which I suggest doing), we can be more efficient by having fewer nodes in multiple hidden layers than tons of nodes in one hidden layer.

The ReLU activation function should take care of this.

ReLU works by fitting short, straight lines to approximate curves. That should be able to create a parabola. You will have performance suffer for inputs with very large absolute values, but we know that models won't be perfect.

I was thinking that one hidden layer could take care of this, but reading about the universal approximation theorem (which I suggest doing), we can be more efficient by having fewer nodes in multiple hidden layers than tons of nodes in one hidden layer.

EDIT

I didn't make this clear three years ago. The universal approximation theorem says that we can approximate on a compact set (on the real line, that means a closed and bounded subset of the number line). Once you go past that bound, all bets are off, which is why I say that you will have performance suffer for inputs with very large absolute values. For a visualization, imagine how an absolute value function ($\vert x\vert = ReLU(x) + ReLU(-x)$) could approximate $y=x^2$ for small numbers, such as $(-1, 1)$, but the approximation is awful for $x=10$, for instance.