These are the rules for the so called "surviving son role" in the draft, explained by snopes.com
So, to clear up the misunderstandings, note that:
You are not exempt from military service simply because you are an only child or an only son. You are exempt only if one of your siblings or parents (mothers are now included as well) has died as a result of military service.
You do not have to be a sole remaining son to meet this requirement - the exemption applies to all remaining sons from qualified families.
This exemption is not in effect during wartime.
(Although all American men are required to register with the Selective Service upon turning 18, the U.S. does not currently have a peacetime draft, having converted to an all-volunteer military in 1973. Whether a draft held while U.S. forces were engaged in military action but war had not officially been declared - as in the current situation - would be considered a "peacetime" draft or a "wartime" draft is debatable; our query to the Selective Service received only the circular answer that since no draft is in effect at this time, we are considered to be at peace.)
As for the "primary caregiver rule", it appears that the fact that you are a primary caregiver DOES NOT exempt you from the draft, but it can buy you a deferment (I'm assuming that the government gives you time to find alternate care).
This information is found on the Selective Service Website.