Such is not the case for my current studio, nor the majority of teaching studios. My studio is only just big enough for the piano itself, plus a student and myself. I have also have a small desk, and three extremely uncomfortable chairs. Realistically though, it's big enough to have three people in the room.
I will usually invite parents in to the first lessons with a new student, especially if the student is under the age of 9. For beginners, having the parent in the lesson is extremely beneficial. Parents can learn alongside their children and emulate my teaching at home to help their kids through practice sessions. Some of my absolute best students are where they are because their parent (usually it's just one) helped them along through the first months of learning piano. There comes a point though, when it's time for one-on-one lessons.
Some parents, as good as their intentions are, are unable to simply sit back and observe. I've had parents interject comments between mine to the student. I've had parents interrupt what I'm saying. I've even had parents feed their children answers to the questions I'm asking. All of these things are examples of what hurts your student's learning process. In the piano studio, there should only be one source of authority, the piano teacher. Because piano teaching is a one-on-one dynamic, to have the child's attention divided between two sources is very confusing. Additionally, because the parent is already the primary authority in the student's life, to have that parent undercut the jurisdiction of the teacher can be a heavy blow, sometimes delivered early in the lesson. This applies to matters of actual piano playing, as well as discipline.
As far as feeding children answers, there's no real excuse. It just doesn't make sense. I use the Socratic method with my students, meaning I will wait as long as it takes (within reason) to get an answer from them. I often don't care if the answer is wrong, I just care that the student thinks about it. To take that process away from the child undercuts the entire process and in a way, wastes more time than if the child stumbled to the conclusion herself. A student (who no longer studies with me) one time had a parent reach across and play the correct note for her as I was trying to lead her to find it. I assumed that the same thing was happening at home, since the child's progress was extremely slow even after a number of months of lessons.
Disciplinary problems can often be solved by having the parents leave lessons as well. This is a really counter-intuitive solution, and one that I never would have come to without observing it myself. I've had students be little monsters with their parents in the room and then fine the next week once the parent is settled in the waiting room. My hypothesis is that children know their own parent's limits and exactly how far they can push it without the parent going off. Children also know that their parent isn't as powerful in public as they are at home. There is no way to chastise a child as severely in the piano teacher's studio as they can in private. I think that students respect the teacher more when the parent isn't there.
Sometimes there are no negative reasons for why the parent should step out. Either the skill level required to understand what's happening is beyond the parent, the parent is not really paying attention (lessons are a great time for parents to catch up on email) or the child would just feel more comfortable on their own. Music lessons are a great way for kids to grow up. They learn responsibility, and take charge of the consequences of their action (or lack of action). Those things are learned best on their own, with the teacher doling out any praise or condemnation on their own. Parents have to do enough of this as it is, and it can mean something very different coming from a person of authority who is not related.
The hard part for the piano teacher is figuring out how to tell the parent when to leave. Luckily, for most of the parents I've had to do this with, it was because the student was advancing and parental supervision was no longer necessary. It was a simple "I think we're ready for one-on-one lessons next week." I often offer an every-other-week model if parents are unwilling to sit out entirely, or invite parents to come in for the last five minutes of every lesson. This allows me to debrief them on anything new, or give instructions on how to practice with them at home.
For kids who are having disciplinary problems, I'll often tell the exasperated parents that one-on-one lessons tend to have a positive effect in this regard. That's usually enough. Some parents are often relieved to sit in the waiting room, because it lets them off the hook a bit. I think there's a bit of guilt in this but there really needn't be.
I've made a little guide below on what to do and what not to do in your student's piano (or other music) lessons. This guide could probably apply to any situation in which a child is receiving personal training.
Dos and Don'ts for your Child's Piano Lessons
DO sit in on your beginning student's first term of lessons. This will help your child feel more comfortable and you will learn how to help them.
DO take notes. I had a father bring a pad of paper with him and he took meticulous notes on the assignments and strategies for helping his child learn. That student is one of my best now and I credit the father heavily. He is also still able to help even though he no longer sits in on the lessons.
DO take this as a chance to observe how your child tackles problems. Think of it as a case study of their entire personality. You are the scientist observing your own child.
DO think of it as your own private pedagogy lesson. Most teachers at a music school now have had many years of experience as well as advanced degrees in their fields.
DO ask questions at the end of the lesson, though try to save them for the end instead of interrupting.
DO give your child benefit of the doubt. They often know the answers that you might not.
DON'T interrupt the lesson, or really interact at all unless prompted by the teacher.
DON'T feed your child answers to questions.
DON'T feel bad about removing yourself if you feel that you're not needed in the lesson. If you're doing emails or checking Facebook in the lesson, go ahead and sit in the waiting room. I recognize that parents are super busy, and if these 30 minutes are your only time to do some personal stuff, by all means go do it.