I don't know for sure, but I think you'll have to work this out with the registrar of your domains, be it Tucows (LP) or someone else. Parking of a domain is usually something taken care by the registrar. Your domain control panel (not LPCP) might be able to handle it, but I'd ask the registrar first.
As for what happens to the files under the domain to be parked, I'd back them up first. Go on the assumption that you will lose the files (if you don't, and still have access to them after the parking, great -- treat it as a bonus). Are these two domains under the same account or different accounts? If different accounts, you may need to close the (now parked) site's account separately, if you don't want to keep paying for it.