It is formally ambiguous, since "had" has a range of meanings. The most likely sense is "The mother gave birth to a boy on that date"

It could mean lots of other things... "had" can mean "eat" so it could mean "they ate a son". But in real life the sense of "They owned, but don't own now" or "They ate" are much more unlikely than the simple interpretation of "gave birth". And there are better (clearer, more tactful) ways to express the other meanings.

It is formally ambiguous, since "had" has a range of meanings. The most likely sense is "The mother gave birth to a boy on that date"

It could mean lots of other things... "had" can mean "eat" so it could mean "they ate a son". But in real life the sense of "They owned, but don't own now" or "They ate" are much more unlikely than the simple interpretation of "gave birth". And there are better (clearer, more tactful) ways to express the other meanings.

Note that meaning "owned in 2010" (which could apply to a car) does not imply much about the present. I had a car in 2010; I still have a car now. You can't deduce from "I had a car" that I do or don't own one now.