They had a son in July 2010.
When you are talking about a time period (year, month etc) in the past, this sentence might mean;
1-On that date in the past, they had a son. (but they don't have a son now) (eg "I had a car" means "I don't have it now"),
2-On that date in the past, the family welcomed a new baby, so now they have a son.
Is this sentence really ambiguous without a context or it simply means number 2?