Yeah, although it implies the father has all the experience and the youth has so little he cannot even find them (the monsters). That would make the vorpal sword most likely his father's. i.e. a fair interpretation. Its not like you find a vorpal sword lying on the side of the road.
Mother: Don't touch that! It might have germs!
Kid: But Mo-oooooooom! It's so shiny and looks so sharp!
Mother: then you'll likely cut yourself! Get away get away! Leave the bijillion dollar vorpal sword to the beggars!