Morroway is an English name, originally a placename, transferred to surname and then, clearly, to given name in your father's case, albeit an extremely rare one. It's rather compound also but the meaning would be 'road where the row of houses on marshy ground are'. Morrow being the row of houses on marshy ground and -way being the road.
Palistine, well, it looks as if it's from the Palestine source, as said. Palistine is from the Roman name for the country, meaning 'land of the Philistines'. Philistines is from the Semitic for 'invader'.
So Morroway Palistine has the rather complicated meaning of 'road where the row of houses on marshy ground are in the land of the invaders'....no wonder your dad didn't tell you ;o)