I remember having a student last summer (2013) who was also taking swimming lessons, kumon, and other summer school activities along with piano. During one of our lessons, she suddenly burst into tears and wailed, "pagod na pagod na ko!" (I'm so tired!). In my alarm I ended up scolding the mom for putting too much on the kid's plate. Probably not the wisest thing for me to do considering I need to earn. XD
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I salute piano parents, I really do. I often look at kids with supportive parents with a touch of envy because I'm the only one in my immediate family who likes to play piano, and enjoys classical music to boot! Which is why whenever any of my kids play at a recital I designate myself as "the applause committee" (clap my hands raw and generally embarrass myself and probably my students, heh), because I know how much encouragement one needs when one is about to play on stage.
But please, piano parents, don't overdo it and exhaust your kids by giving them too much to do. Keep in mind that piano playing is both a physical and mental activity.