I think Crassus's eldest son, Publius, died before his father at Carrhae. He led his father's Gallic cavalry wing (perhaps confirming that Publius had at some stage been in Gaul with Caesar) and after being heavily defeated and cut off from his base, Publius and his commanders committed suicide instead of fighting on in an already lost battle.

There is even less about Crassus's second son, Marcus. Anything I have read on him say that he too died at Carrhae.

I would suggest that because the two sons died fighting with their father that their loyalties lay with their family