Welcome to the club....parents like that come around every so often...you just have to learn how to handle them
First, you should allow your classroom to be an open book and allow for the parents to see everything your doing. Parents want to know a) there child is learning something, b) the teacher can teach, c) things they can do to help their child at home. Nobody likes to be judged but we all do it. The kids judge you, parents judge you, and so do other teachers. You want to be so on top of your game it does not matter who comes in your room. You have a plan and they just happen to witness it.
As much as I hate Fur Elise and Canon in D, there is a place for them because they are recognizable, kids (adults too) love them, and they can be used as a benchmark for level. You have to kind of play mind games with students and parents a little to steer them away from pieces. If the parent insist you teach them and you know it is beyond their level go along with them. Ask them basic questions the would not now such as time signature, grace, notes, rhythm, expression meanings in the music. The minute they say they do not know , suggest a piece or book that would help them understand their goal. Explain they will learn these concepts to help us play the piece that is being forced on us and will return to it shortly. Whether you actually do is up to you but that way they realize they learn the basics so they can play the challenging piece, you are willing to challenge them but want them to be successful, and the parent feels validated because you listened to her and acknowledged her need to be taking seriously ( yes...you have to meet the needs of the parent as well).
Last thing with the make-ups, you want to honor your commitments no matter what. Parents do not want to hear their child has the attention span of gnat. No matter what you say to them, if you let their kid out early, they think your lazy. Sorry but its the truth because changes our they never had to teach a kid for an hour so I feel your pain. If a kid looks bored in anyway, their telling your teaching pace is too slow, your talking too much, or you need to give them an activity. Make them do something active the list goes on...counting, saying note names, playing with eyes closed, echoing, repeating, taping . You want the kid actively involved to keep the attention up and to ensure they understand what to do when they get home.
I am completely convinced serious pianist come from inwardly enjoying music and being able to be successful at it. If you can make them enjoy and be successful you will plant a seed that will continue to grow beyond your knowledge. I have seen many students go from knowing nothing to playing Fur Elise within a couple of months because they love to practice the piano not because they teacher pushed them to practice. With older teens and adults it works to have someone pushing you to get better but with kids you really want to spark their imagination even if their on the older teir ( 9-10). If you can connect the music to place, a type of sound, an animal they like, a color, and combine them with a parent that helps them find time to practice they blossom and be your best students. T