I do feel your frustration about your DD getting injured during a foul. My DD suffered a few fractures while being fouled. The referees got the calls correct. One yellow card and two fouls. The only injury the referee got the call wrong was a concussion while she was playing in college. Should have been a yellow card and he called nothing. He got dropped as college referee the following season due to multiple complaints from college coaches and poor assessment.
Referees have criteria we must follow when calling fouls and issuing yellow and red cards. The fact that a player gets injured due to a foul does not push it from a simple careless foul to a red card. The only criteria I can think of that would result in a red card for a push from behind would be if the push used excessive force to the back or the head. Almost all pushing from behind will result in either a foul or foul-yellow card. I am not sure what constitutes a “dirty” push, but “dirty” is not listed in any foul criteria in the HS rules or LOTG. Can you explain what you mean by “dirty” such as what part of the body (hips, lower back, upper back, shoulders, neck, head) was pushed, position on the field, number of defenders between player and the goal.
A parent that comes uninvited out of the stands and onto the field is a problem. Referees do not know if that parent just wants to help their kid, physically or verbally attack the player who committed the foul or physical or verbally attack the referee. I have seen all of those scenarios occur. The coach is responsible for controlling their spectator. The coach(s) and athletic trainer(s) should be the nay ones entering the field to care for an injured player.