I don't really buy the argument that the NFL has so much more parity than any other league but I suppose that depends on your definition of parity. In most leagues around the world, the team on top can be beaten by the team on the bottom if the top team doesn't show up for that game. However, the top teams usually stay the top teams and the bottom teams stay on the bottom for a few seasons barring influxes of cash, change in management style, the like.
In the case of the EPL, the Big 4 clubs have been pretty stable over the past several years. Arsenal have had Wenger since the mid to late 90s, Sir Alex was at Man U since the mid 80s, Abramovich bought Chelsea about 10 years ago and they've won a glut of trophies since, and Liverpool has been at the top for a while too (though I believe they're a little less "stable" when it comes to ownership and coaches, might be wrong here). The NFL is more or less the same: the Patriots have had the same coach and QB for 10+ years, the Steelers have had a QB and very stable coaching, and the Colts as well have been very consistent with Manning as they're QB for a long time.
The teams in the middle beat each other up, and the teams at the bottom don't usually improve that much year to year. Over time, the giants will fall if their stability takes a hit in some way, and the teams further down the table will most likely hit on something and start working their way up. Case in point, Man City was relegated in 2001 and they're now in the top 3. I think there may be a perception of parity in the NFL due to the coverage here about every game while the EPL largely only gets press in the USA for the Big 4 teams unless you follow the league.