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Solution:

The answer is $\mathbf{(E)}$. Observe that the expression $Q(x) = P(x) - x^{42} P(1 / x)$ is a polynomial, since the monomial $x^{42}$ simplifies the denominator of $P(1 / x)$. Moreover, it has degree at most 42, and if $r$ is one of the first 21 odd prime numbers, $Q(x)$ vanishes at $r$ and at $1 / r$: indeed, we have
$$Q(r) = P(r) - r^{42} P(1 / r) = 0 \quad Q(1 / r) = P(1 / r) - (1 / r)^{42} P(r) = 0,$$
where we used the fact that $P(r) = P(1 / r) = 0$ by hypothesis. Finally, $Q(x)$ vanishes at 1, because $Q(1) = P(1) - P(1) = 0$. Since $Q(x)$ vanishes at at least 43 distinct values of $x$ but has degree at most 42 we obtain that $Q(x)$ is the constant polynomial 0, hence $P(x) = x^{42} P(1 / x)$ holds for every $x$. We therefore have
$$\frac{P(2)}{P(1 / 2)} = \frac{2^{42} P(1 / 2)}{P(1 / 2)} = 2^{42} = 4^{21}.$$