Staff travel kicks in usually after 6 or 12 months service, and should be you and your folks for travel on BA, and all the airlines that BA has an interline agreement with.
Your folks should be able to travel on BA, and on some other airlines without you.
As for allowances and flight pay Etc - I am not up to speed on the new BA contracts sorry - No doubt you will get an answer here soon.
Good luck with your decision.
What about your NHS pension?