Can my parents use frequent flyer miles to pay for my ticket?

No. Frequent flyer mile tickets typically do not meet Rotary requirements. Usually the airlines allot for only a few seats on each plane for frequent flier tickets, and these seats sell out quickly. If it became necessary for you to come home in an emergency, you may not be able to find space on a plane which would cause you to purchase a (very expensive) one-way ticket home.