My dad is being all douchey, hes saying that LoL uses more internet bandwidth if i play on high specs, is this true?

Yes.
Because the game is more enjoyable on high specs therefore you play more therefore it uses more bandwidth.

I would play it the same anyway .__., but is dad's theory correct?

I prefer not to start a war with your dad

if i would program a game, then i would only submit the coordinates of the chars/items/etc over the network. if you send the whole screen, then games would NEVER be that fast.
why would other programmers do anything different? what are the reasons? maybe your game does 'dynamically' downloading graphics? i don't know LoL, but 99.99% of games won't use more bandwidth.
i heard d3 downloads 'dynamically' things, for the reason that people can't build an offline version... but i think they didn't take the effort to download graphics in different resolutions.

-> your dad is obviously wrong

This is correct. The same data is sent through the server, higher specs just change the way your computer handles the data