TL/DR Version: NFL officiating is really hard, and games are limited - refs can't get better without experience/more games, and there aren't any more. Possible fixes: less ref crews doing multiple games/week (more experience), rules changes to make things more black-and-white and less 'judgment call', and expand replay opportunities to get calls correct...

LONG POST:

It doesn't help that football is, by far, the most difficult game to officiate, and that this most difficult game to officiate also has, by far, the most limited opportunities for officials to practice/get better at it.

For comparison's sake: the NBA has 1,230 regular season games... MLB has a whopping 2,430 regular season games... NHL 1,312... and the NFL a paltry 272.

Add to that, the way they are instructing officials to call the NFL is constantly changing, and it seems like the most argued-about calls are either judgment calls or "things you could call on every play", so the refs are constantly in a position of having to decide "should I let this one go even though by letter of the rule, it's a foul".

I really think that last is the biggest problem. Think about it. You're a referee. You see offensive holding on every... single... play. But you can't throw a flag on every play - if you do, you'll ruin the game (and probably get canned). So you have exactly one second (sometimes less) to decide whether THIS PLAY warrants a flag, while THE LAST PLAY didn't. Every human being is going to see that differently. Which leads to the other significant issue in NFL officiating. You now have official crews who are well known for being flag-happy, and those who are not. Well, they're human. They're going to see it differently. But as a player (or a team), it's really hard to play the game when one week you get away with stuff, but the next week you do the exact same stuff, and it's flags every third play.

How to fix it?

One thing that might help might be to spread the games out over more days. You have 16 games per week (less if there are byes). If you played 2 on Thursday, 2 on Friday, 5 on Saturday, 5 on Sunday, and 2 on Monday, that gets you to your 16. In that case, if you could use the Thursday crews on Saturday, your Friday crews on Sunday, and two of your Saturday group also did Monday, that's six officiating crews getting an extra game of experience per week. That also gets you down to needing 10 total officiating crews (rather than 16) per week - weeds out the six worst crews. Point being, whatever you can do to get these guys more opportunities to officiate. I am convinced that the only way to get good at officiating NFL football is to do it - and the speed of the game and the way the NFL game is played is so far and away different from the college (or minor-league) game that I'm not sure what's to be gained by doing those other things.

Another needed fix is to revamp the rule book in such a way that more things are obviously enforceable by 'letter of the law' - take away some of the 'should I throw a flag' element. To whatever extent possible, make the rules be 'either it's a flag, or it's not' - black and white. And yes, there would be an adjustment period where a LOT of flags would be thrown, but hopefully it wouldn't be too terribly long before teams learned how to play under the 'new' rules/officiating standards.

Third: expand the use of instant replay. College uses it to review plays for 'targeting' fouls - it's quick, and they usually get it right. And the NFL is already using some sort of 'quick' replay where the officials have overturned an initial call without a challenge or anything. This could be expanded. Oh, and along with that, double the number of coaches' challenges, and make ANY call or non-call open for challenge. If the coach thinks there's an obvious PI or face mask that went uncalled (or the hands-to-the-face that Parsons constantly gets hit with but is never called), they should be able to throw the challenge flag. Or, make it like the NBA - you get one challenge per half, but as long as you're right, you keep it and can keep using it. But if you're wrong (call not overturned), then you lose it. That might be a nice balance in that it would limit it to the most crucial (or blatantly obvious) plays...