There is no rule that mentions a video barrier or video being required.

I mean I accidentally spammed them, but still.

Don't worry about the spammed submissions, sometimes websites malfunction.

I added a line to the rules that signifies that only submissions with videos will get accepted. I just don't see a reason to have runs on a leaderboard without any proof.

You can still run the game and if you want your run to appear on the leaderboard you could use a webcam or something like that if a capture card is too expensive. You can get pretty decent quality with webcams nowadays.

The problem for me could be not being allowed to record videos.

If I understand it correctly, why aren't you allowed to record videos?

I am still a child, and I believe my parents wouldn't want me to.

Just ask them and tell them that you really want to do that.