  We present lower bounds for the orbit length of reduction modulo primes of
parametric polynomial dynamical systems defined over the integers, under a
suitable hypothesis on its set of preperiodic points over $\mathbb C$. Applying
recent results of Baker and DeMarco~(2011) and of Ghioca, Krieger, Nguyen and
Ye~(2017), we obtain explicit families of parametric polynomials and initial
points such that the reductions modulo primes have long orbits, for all but a
finite number of values of the parameters. This generalizes a previous lower
bound due to Chang~(2015). As a by-product, we also slighly improve a result of
Silverman~(2008) and recover a result of Akbary and Ghioca~(2009) as special
extreme cases of our estimates.
