CIC allows it, and I actually did that, it's called dual intent:
http://www.cic.gc.ca/english/resources/tools/temp/visa/dual.asp
The Come to Canada tool is only a wizard for online applications, it's not the actual ministerial instructions.
I for one applied for a TRV while my PR application was in progress. I answered the questionnaire as if I was only applying for a TRV, I even used the same MyCIC account and UCI number as my Express Entry app, and there was no conflict at all: My PR application was approved 2 weeks after my TRV was stamped. I got the GCMS notes for both applications, and CIC was perfectly aware of my parallel applications.
I reused my PR medical for my TRV medical (although not required). So if you do upfront medicals for Express Entry, you can submit the same one for your study permit.