If you're going to use the possessive apostrophe+s then your first example should be "Tom's and Linda's father".
"Tom and Linda's father" is ambiguous and could mean two different people:
2) Linda's father
"Tom's and Linda's father" is not ambiguous. It's clear that one man is the father of both Tom and Linda. If they had different fathers, it would read "Tom's and Linda's fathers".
- For Teachers