Something similar, but different: I was closely involved with a soccer association that didn't keep score in their "in house" games, and the "no score" rule was made clear to everyone at the beginning of every season. Nonetheless, at the end of one season, some over-zealous parent took it upon themselves to send a picture of their child's team to the local newspaper (and they published it), along with a small write-up that boasted that their team was "undefeated for two seasons".
Let's just say that the association's board-of-directors did not look too kindly on the matter.
The reason that they don't keep score in the little kid leagues isn't because of the kids. The kids all know the score. At the end of the game they know if they've won or lost or not. They don't officially keep score because way too large a percentage of parents are arseholes who would make the games all about winning and not about the kids having fun learning the game, which is what it should be about at that age.
Back when my son was playing in U7 or U8 we got there early for a game. We were the first ones on my sons team there and there were a couple kids from the other team there. So, as kids will do, they just started kicking the ball around together and taking shots on goal. I was talking with one of the other dads when the coach for the other team got there. He tells us that before the game started that night they were going to make an announcement to the parents basically telling everyone that they need to keep things calm during the game and leave the refs alone. Why? Because at a game the night before a parent was pissed off at the ref for missing a call and he went out on the field to confront the ref. Oh, yeah, did I mention, at that age group they used older kids as refs to help the kids learn the game better?
Think about that. A parent was so pissed off that a 13 year old kid (and I don't know exactly how old he was, but close enough) who was refereeing a game missed a call that he went out on the field and confronted the kid. At a U8 game.
If the parents had the same attitude about these games as the 7 and 8 year olds did there wouldn't be any problems.