Why the parents, because learning a new task takes work and it may become the parents' job to help their child play piano each day. My job is to make playing look easy and break down each step into small enough parts to grasp. The parent can and does listen and encourages way more than I do seeing the student once a week. I really want to make sure that even when the student gets frustrated or bored that we have an open line of communication. Policies are covered, to make sure that we all start on the same page with the same expectations. Type of instrument is covered, I wouldn't ask to join ice skating without skates or with double bladed skates, but so many parents want to try out piano without having an instrument at home.
What am I looking for in the student? Actually not necessarily musicality or artistic bends. Most importantly am looking and listening to see how they follow directions, and if the students' schedule has space for another activity. While playing and improvising on the piano I'm asking the student to follow my lead on high low, bouncy, smooth, or loud and soft. The student will be asked to draw a picture containing 3 items. This may seem like a strange request, but it really isn't. During our time together, I may have to ask the student to fix 2-3 things in a short piece which they will need to remember the rest of the week. Asking them to remember such things is harder for some students than others.This helps pick out lesson materials. I don't automatically assign the same books to each new student as a matter of fact this year I have students working out of 3 different method series some containing multiple tracks within the series. As for wondering about other activities, I do believe that especially young students need time to play and be creative on their own, so if I hear your child is an activity 3 times a week, swimming lessons, and now you'd like to also add piano, I may ask when each day will the student be able to sit down to play piano?
Piano can teach many great things however it needs to be something that isn't expected to learned overnight and will be frustrating sometimes. -- My mom can't even tell you the number of times I wanted the throw the metronome out the window.