Can a male yorkie poo mate with a female beagle?

In general, you should not allow any dog to breed with its dam or sire. Such close breeding can cause behavioral, conformational, and health issues. Basically, the genes of the puppies will be from a smaller pool, ultimately receiving 3/4 of their genes from the mother. Although you could have healthy puppies, the chances of problems arising are drastically increased because of the possibilities for these alleles to appear being multiplied.

Short answer, no, it really is NOT okay to allow a puppy to breed its mother.