Wikipedia lists many honorifics for one's own family, but I don't see any for one's own (living) mother or father. How would an adult write the equivalent of "Dear Mom" or "Dear Dad" in an email? Does "亲爱的妈妈" sound childish?
Screenshot of an example letter written to one's parents from Using Chinese by Yvonne Li Walls and Jan W. Walls:
Using Chinese is a very useful book, it covers many topics text book usually skip, like proper names, geographic names, numerals, fractions, formal and informal letters, names of holidays, and many useful expressions (invitations, apologies). I am not affiliated with it, I just simply think it is a book that covers many questions that pop up during learning Chinese and it saves you from having to research them on your own.
You are always a child in mother's eyes,So ... childish is acceptable .
You can also use it to your close friend :
Email nowadays is quite informal, like sms. I never use any salutation to family members
I write paper birthday cards to them in which I start with 爸爸 or 妈妈 (depending on whose birthday it is)
To use “亲爱的爸爸” or "亲爱的妈妈" has nothing to do with being childish. In Chinese culture, we don't directly express our love for parents,especially people born before 1980.