Answer:
Whether "your father and I" or "your father and me" is the correct construction depends entirely on how it is used in the sentence. "Your father and I" would be correct only where the use of "I" alone would be correct, and "your father and me" is correct only where the use of "me" alone would be correct.
"Your father and I" is correct when used as the subject of a sentence. For example, "Your father and I have decided that you will be grounded for the next month."
"Your father and me" is correct when used as the object of a sentence. For example, "Look at the beautiful gift the city gave to your father and me."