Is it permissible for us to distribute the father's Udhiyah from the cattle and keep the mother's Udhiyah from the sheep whole, and does that affect the reward?

If your mother did not give away any part of her sacrificial animal (the sheep) in charity, and instead relied on your giving from the seventh share of a cow, this is an error. This is because it is obligatory for the one offering the sacrifice to give away some of their sacrificial animal in charity, even if it is a small amount that fulfills the meaning, based on the Almighty's saying: (So eat from it and feed the distressed and the poor). As for exchanging sacrificial meat, such as a father giving meat from a cow to the mother for her to give in charity in exchange for her taking meat from the sheep, this is a sale, and selling sacrificial meat is not permissible.