You have to reach back to pre-deregulation (pre-1978) to find the "much more expensive" inflation-adjusted fares the OP's dad was talking about. In the 1990s I figured $400 was about the target fair price for a transcon RT; today you can find scattered deals operating at weird hours in the $300s, but $500 or $550 is a more reasonable target. In the 1980s and early '90s there were plenty of $29 and $39 one-way fares up and down the northeast US thanks to New York Air, PeoplExpres, Air Florida, Presidential Airways, and the other destabilizing entrants. Today those same routes are flown by RJs or turboprops instead of mainline jets, reliability and service are worse, and the fare is liable to be $200 or $300, not $29 or $39.

International fares are more expensive today. I too flew PeoplExpress EWR-LGW a couple of times @ $298 RT, but also never paid more than $1,000 for a transatlantic ticket in economy on NW, TW, etc. $600-$800 was more like it. Today $1,000 is pretty much the floor price thanks to less competition, more fees, and higher taxes.