You do have to supply some sort of evidence that you have accomidation in the UK and that you are authorized to live there. If it is let, then a letter from the landlord is enough, if it is mortgaged, generally the person who holds the mortgage had to provide proof that the property is mortgaged so prove they can give permission for you to live there.
Also without the payslips, they may question if the parents can really earmark the money for support.
The process is somewhat subjective, but if you look at it from the ECOs point of view, they want to make sure a) you legally have some place to live and b) you can live in the UK without government support. They want and often require more then just someones word that it will be ok.