What your talking about here is called wheeling, I used it a lot when I first started gambling and I still use it today in a slightly different fashion. Back then I would try and find one game that I would parlay with 10-20 other picks in two game parlays. If you hit your wheel game you need to win only around 35% of your bets to turn a profit. It was a great when I was betting 100 bucks a pick but as time went on and I was up to 300 bucks a pick losing (the wheel) became to stressful.
What I do now in the NFL to increase my roi is pick a college football game on the money line, something in the -7 range, I'll use -335 here for an example so you parlay the -335 to all your -110 NFL picks. That particular 2 game parlay pays +148 and it's not that hard to see the upside. Downside is you miss your wheel game and lose every bet, but it's not that tough to find a -335 college winner on Saturday to use on Sunday. Next problem is being shown the door I'm afraid....
I also believe trytry is right about the software issue and the line staying put but it's a limit thing more than anything else,2500 Canadian is the max at Pinnacle for instance on a parlay and 37,000 for straight bets so most of the big players don't bother and that's why the lines don't move.