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In this paper, we make partial progress on a function field version of the dynamical uniform boundedness conjecture for certain one-dimensional families 
${\mathcal{F}}$ of polynomial maps, such as the family 
$f_{c}(x)=x^{m}+c$, where 
$m\geq 2$. We do this by making use of the dynatomic modular curves 
$Y_{1}(n)$ (respectively 
$Y_{0}(n)$) which parametrize maps 
$f$ in 
${\mathcal{F}}$ together with a point (respectively orbit) of period 
$n$ for 
$f$. The key point in our strategy is to study the set of primes 
$p$ for which the reduction of 
$Y_{1}(n)$ modulo 
$p$ fails to be smooth or irreducible. Morton gave an algorithm to construct, for each 
$n$, a discriminant 
$D_{n}$ whose list of prime factors contains all the primes of bad reduction for 
$Y_{1}(n)$. In this paper, we refine and strengthen Morton’s results. Specifically, we exhibit two criteria on a prime 
$p$ dividing 
$D_{n}$: one guarantees that 
$p$ is in fact a prime of bad reduction for 
$Y_{1}(n)$, yet this same criterion implies that 
$Y_{0}(n)$ is geometrically irreducible. The other guarantees that the reduction of 
$Y_{1}(n)$ modulo 
$p$ is actually smooth. As an application of the second criterion, we extend results of Morton, Flynn, Poonen, Schaefer, and Stoll by giving new examples of good reduction of 
$Y_{1}(n)$ for several primes dividing 
$D_{n}$ when 
$n=7,8,11$, and 
$f_{c}(x)=x^{2}+c$. The proofs involve a blend of arithmetic and complex dynamics, reduction theory for curves, ramification theory, and the combinatorics of the Mandelbrot set.
