Plot VERY similar to The Outer Limits Episode 1997
Just watched episode "Last Supper" starring Fred Savage as the son, Peter Onorati as dad.
Although the reason the woman is immortal is different (in OL, it was not clear but something due to immunity she may have developed to the Black Plague), the rest of plot is similar. Dad meets girl, they have romantic involvement but go their separate ways (in OL he saves her from evil military experiments, but he is married and stays with wife).
But most of the story takes place when son brings girl home to meet mom and dad. Dad recognizes girl. Girl acknowledges who she is. Son is confused. Dad points son in right direction. Son and girl live happily ever after.
Oh...in OL episode, the evil experimenter meets an ugly end.