First of all - congratulations!
As far as medicals are concerned, you can undergo your tests here while your parents in India. However, with regards to passports, you may have to mention on the covering letter (of passport request) about your situation and that once the application is finalised you would submit your passports. This is just a suggestion, not sure if CIC would accept it.
Edit: You cannot send your passport to India. As per Indian laws, it is illegal to send passport overseas by mail. But if you have someone going there, that`s a different issue. Then again, you`d have to worry about landing formalities.