  We develop techniques for using compactifications of Hurwitz spaces to study
families of rational maps $\mathbb{P}^1\to\mathbb{P}^1$ defined by critical
orbit relations. We apply these techniques in two settings: We show that the
parameter space $\mathrm{Per}_{d,n}$ of degree-$d$ bicritical maps with a
marked 4-periodic critical point is a $d^2$-punctured Riemann surface of genus
$\frac{(d-1)(d-2)}{2}$. We also show that the parameter space
$\mathrm{Per}_{2,5}$ of degree-2 rational maps with a marked 5-periodic
critical point is a 10-punctured elliptic curve, and we identify its
isomorphism class over $\mathbb{Q}$. We carry out an experimental study of the
interaction between dynamically defined points of $\mathrm{Per}_{2,5}$ (such as
PCF points or punctures) and the group structure of the underlying elliptic
curve.
