formal policy - Generally not. Firstly the 911 system (I'm a 911 volly) is inconsistent at best in most areas, while it doesn't hurt for them to request the "system" most won't do it until they really need to.
Your father's case -
Either they already knew that few members were around (a weekend in the summer perhaps) or more likely they tried and failed to get any units to respond.
Formal policy or not, practically speaking in Brighton Beach, Coney Island and neighborhoods not covered by regular members they will call 911 after trying 4 or 5 times to get a first responder or units for a bus. Once the system is activated they will ask for a first responder to go until the system gets there. I have never heard them call 911 for any calls in Queens. Everything is covered in Queens no matter the ETA.
As far as mutual aid goes in the Catskills they will respond to calls for mutual aid, mainly medic calls, but I have yet to hear them ask for mutual aid other than the chopper. There was just a call in Kerhonkson yesterday and they dispatched a medic from Fallsburg, a 40 minute drive without lights and sirens, instead of calling for mutual aid mobile medic.