I have a few from over the years:
1. Got an email a month after track season forwarded to me by my AD. One of the moms had complained to him that I only let her daughter run in two meets that year and I never gave her any recognition or encouragement. She ran every meet except for the conference and regionals (girl was a 6:45 miler and last place at those meets was usually 6:10-6:15). I also put together a 4 x 8 relay for an end of the year freshman meet (got a 400 runner to move up) so she could run in that meet even though it was an extra event for the other athletes in the relay. Mom even thanked me at the time for doing it. AD backed me up when I told him I explained this to him. Turns out the same mom had called the principal on four of her daughter's teachers for various reasons.
2. For our first track meet of the season, I had a distance runner who only wanted to run the mile each meet. I thought it would be better for her to experience some other events that would help her become a better all around distance runner. For the first meet, I put her in the 800 and 4 x 4 and said would run the mile the next meet. At the meet the next day, her dad makes a comment in front of a bunch of other parents while I am close by asking why I don't put the kids in their best events (I had never met the guy, but I guessed whose dad he was). Turns out the XC coach (who had applied for my track job) was encouraging dissent. Dad didn't know what to say later in the season when my coaching strategy paid off and his daughter had taken 20 second off her best mile time and medaled at regionals.
3. Mom of one of my freshman boy distance runners called the AD's office phone at night and left anonymous messages questioning why her son wasn't allowed to run the maximum four events each meet since she had paid his participation fee. I tried to explain that it wasn't in her son's best interest (especially since he had come into the season injured from basketball) to run that many events. She kept complaining to the AD and the other parents. The next year, there was a new coach who ran him in 3-4 races each meet. He was never better than his freshman year.
4. Weirdest one had to be a girl on my XC/Track teams whose parents (who appeared to have lots of money) would show up in the middle of practice a couple of times a season and say they had to leave immediately for "vacation". They would be gone for a couple of days (during the week) and then come back. The daughter never told the other girls where they went. At a dinner the night before a meet out of town, I was sitting next to the dad and tried to make conversation by asking him what he did for a living. He completely dodged the question and seemed insulted that I asked (I quickly changed the subject). A couple of other parents later wondered why he had been so evasive.
There are more, but those are the ones that came to mind when I ready the thread. Glad to see I'm not the only who dealt with wacky parents over the years.