I’m curious to hear what comes of this from the OP.
I run a small but successful program where most of my business is from referrals and many of my students have been with me 6+ years (I’ve only been teaching in town for 8 years). I’ve never had one down right rude, but I’ve had a few with some anger issues or problems. Usually I’ve been successful helping students, but some I’ve referred to other local coaches because they simply did not fit with my program.
Here is my take, worth every penny you paid for it 
A: Easy answer - drop the child as a client, keep brother, explain to parent that you simply cannot keep that attitude around. Explain that you would be willing to revisit in future, but at this time it cannot continue.
B: Make-Work Answer: Sit down with parent at scheduled meeting or phone call, no children. Explain behaviour including examples, be supportive of parent but firm that behaviour must change and quickly. Offer potential solution:
-child in private lessons only, shorter riding time and fully supervised by you at all time while getting ready and putting away. Child is not to be left alone on property and is not to linger before or after lessons. Threats mean there will be NO alone time with any horse.
-sit down conversation with child and parent in room, explaining in plain terms the behaviour that you have seen from child that has to stop, and recommend the behaviour that you should be seeing instead. Give examples of scenarios, language and actions so an 8 year old can understand, and be sure parent is ok but the conversation has to be led by you. this helps child see that parent respects you.
-reward the positives you see, and explain why they are positive "you just gave Spot a pat, that was nice of you’ or ‘you did a nice job asking Spot to back up properly. Good job correcting him when he tried to walk past the mounting block’ etc. I’d guess she is trying to show superiority as a way to build up her confidence, which is probably a smokescreen for a little girl who is afraid and lacks self confidence. Give her measurable goals like riding a pole course, telling the correct posting diagonal 3 out of 5 tries, tacking up on her own etc and let her know she did a good job. Then when she demonstrates ability - let her do it. Supervise tacking up but do not help. Supervise her getting the pony from the field, but do not help, etc. Tell her ‘I know you can do it, and I know its hard. But keep trying and I bet you can do it on your own’. Put poles down in the arena, have her ride a pole course you make up, then have her make up her own. It will start to give her confidence in actual skills vs the All Talk approach she seems to have going on now
-set goals that are measurable like the child using please/thank you, patting the pony, asking you for help etc and after you see positive behaviour in so many lessons, allow child to ride in group lesson again. Consider alternating between group and private to be sure behaviour improves in both settings as an ‘audience’ might be a factor.
If parents are not the cause and are on board with the solutions, it shouldn’t be too big an issue to correct and the parents should be grateful for your help. But that is assuming they are not part of the problem…
