The Blue Lagoon
(1980)

When they realize that little Paddy has swallowed several berries from the sleep-inducing plant, Richard (Christopher Atkins) and Emmeline (Brooke Shields), overcome by grief and seeing that their son is doomed, decide to do the same thing. 

Eventually, they fall asleep too, and shortly after, a ship rescues them.  Arthur (William Daniels) turns out to be among the crew.

Thanks Dr. Mat!
(who also sent us the 1991 Sequel - RETURN TO THE BLUE LAGOON)