Does Turbo Tax electronically file either or both 1041 and PA41 for Pennsylvania Simple Testamentary Trust that I am trustee of?

for 1041 you need desktop Turbotax Business. There is no online program for trusts. Technically it's you who e-files the 1041 using TurboTax by clicking the submit icon. someone else will have to answer about PA. 

Thank-you for answering that I can file the 1041 electronically through Desktop Turbo Tax.  I'm hoping someone can answer that I will also be able to file the PA 41 electronically through Turbo Tax.  Or is Pennsylvania a state that Turbo Tax cannot e-file PA 41?

You are correct that Form PA-41 must be printed and mailed. According to the TurboTax Business State forms availability table, all Pennsylvania trust forms must be mailed.