Is it obligatory to fulfill a vow made for the birth of a mother to a cow owned by the father, which contributes to expenses?

This vow is not valid, and nothing is required from it, because it is a vow concerning something one does not own. The Prophet, peace and blessings be upon him, said: "A man is not bound by a vow concerning what he does not own," and in another wording: "There is no fulfillment for a vow involving disobedience, nor for what a servant does not own."