Brainteaser 1633: New Year’s resolution

From The Sunday Times, 26th December 1993 [link]

My New Year’s resolution is that this year I will remember my father’s birthday. To remind me I devised this formula: if I take the factorial of the number on the calendar representing the month in which he was born (1 for January to 12 for December) and multiply this by the square of the day of the month of his birth (1 to 31), the four digit result is the year when he was born.

[Note, for example, that “6 factorial” is 6×5×4×3×2×1 = 720.]

I then realised that this is no use for determining his birthday as even my own date of birth fits the formula.

Can you tell me what is my father’s date of birth?

This puzzle was included in the book Brainteasers (2002, edited by Victor Bryant) under the title of “Birthday resolution”. The text was changed slightly, but the puzzle remains the same.