Parents can play an important role supporting their child's piano lessons and their piano teacher.
However, about a third of parents do too much and try to take over responsibility for their child's progress, which hinders their child's ability to develop positive learning skills on their own.
In contrast, about a third of parents don't do enough, and fail to support and can get in the way of their child's learning.
The remaining 1/3 gets it right.
These parents have learned to work together with the piano teacher to support their child's piano journey.
Parents who are not getting it right fail to effectively support their children in piano lessons because they either take too much responsibility for their child's success or not enough.
Getting this balance is important because it sets boundaries that create a positive learning environment where children can learn from their mistakes within limits.
Parents' attitudes about your child's piano lessons matters for setting boundaries and teaching learning values.
For example, Regarding your child's piano progress, do you

(a) feel personally embarrassed if your child is not progressing as fast as you think they should; (b) want your child to do their best and learn how to achieve their goals; (c) feel that it doesn't really matter how your child is doing, as lessons aren't that important anyway.

If you chose (a), you may be taking TOO much responsibility for your child's success and creating dependency instead of fostering your child's ownership and responsibility of their lessons.
If you find yourself answering (c), you may well be abdicating part of your responsibility as a parent and may be missing out on a wonderful way to encourage your child's success.
If you have answered (b), relax.
You've got the right attitude and are encouraging your child while helping them take their share of responsibility for their piano success.
Keep it up!