= Solution
A point $y$ is a <regular value> of $f$ when every derivative $D_xf$ for $x$ in $f^{-1}(y)$ is surjective. <Sard theorem> says that the critical values of a smooth map have measure zero. The stack-of-records theorem says that the inverse image of a regular value is a smooth submanifold of codimension $\dim Y$.
Maps $f$ and $g$ are smoothly <homotopy>[homotopic] when a smooth $F:X\times[0,1]\to Y$ has endpoint maps $f$ and $g$. Perturb such a homotopy relative to its ends so that it is transverse to $y$. Then $F^{-1}(y)$ is a compact one-dimensional manifold whose boundary is $f^{-1}(y)\sqcup g^{-1}(y)$. Every compact one-dimensional manifold has an even number of boundary points, proving the parity formula.
The hypothesis on $U$ says that the <degree modulo two> of $h$ is one. Suppose no $h(x)$ and $h(-x)$ were antipodal. Then
$$
k(x)={h(x)+h(-x)\over |h(x)+h(-x)|}
$$
would be defined, even, and homotopic to $h$ by normalized straight-line interpolation. For a regular value, the fibres of the even map $k$ occur in antipodal pairs, so its degree modulo two is zero. Homotopy invariance gives a contradiction. Hence some antipodal pair is mapped to an antipodal pair.
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