I have a data set composed of apnea counts....

The data are no where near parametric.

It can be more helpful to think about the specific characteristics of a variable, rather than just trying to broadly classify it as parametric or non-parametric. In your case your response variable is a count variable, so rather than hunting for a non-parametric test, it might be more helpful to find a model specifically suited for count data.

(I'm assuming that your apnea counts are counts of the number of episodes of apnea for each mouse?)

Two count models that might be useful are the Poisson and the negative binomial. You could still do factorial type analysis, but where an ANOVA would assume a normal distribution of the response variable within each condition, you would instead assume a Poisson or negative binomial distribution. Note that these are parametric models, but the parametric assumptions being made are ones that are plausible for count data.