The pooling of litters for an individual set of parents seems reasonable to me. If the data could be put into a kxk contingency table and the male mice could be naturally paired with female mice a generalization of Mc Nemar's test could be given. This is a generalization of the the McNemar test for K>2. This paper points out the generalization which can be credited to both Stuart in a cited 1955 paper and maxwell in a 1970 paper. The authors of this article show how to implement the procedure in SAS and provide some examples.
The three relevant papers are:
 Maxwell A.E. (1970). Comparing the classification of subjects by two independent judges. British Journal of Psychiatry, 116: 651 - 655.
 McNemar Q. (1947). Note on the sampling error of the difference between correlated proportions or percentages. Psychometrika, 12: 153 - 157.
 Stuart A. (1955). A Test for Homogeneity of the Marginal Distributions in a Two-Way Classification. Biometrika, 42: 412 - 416.
I gave this information in my original answer but pairing is by parents and not by the offspring mice. So given this information it appears these test do not apply.
However the design looks like it could represent a series of kx2 contingency tables where each table represents columns for males and females where k is the number of litters in each group and the cells represent the number of males in the litter or females in the litter depending on the colum the cell belongs to. There is one such table for each set of parents. The tables can be looked upon as a stratification of the data stratified by parents. If the value of k is the same for each set of parents (all sets of parents have the same number of litters) there is a test that compares the distribution of male and female mice taking account for the stratification. The test is a generalization of the Cochran-Mantel-Haenszel test describe here for example.
If you could control or have controlled the experiment so that each parent pair produce the same number of litters you can use this test. If not I do not know yet if there is something similar for unequal stratification groups (counts here being number of litters for the ith parent group).