Ownership is determined by the legal owner on the registration.  If there is an additional oral agreement (as you stated there is "no paperwork"), that is an issue for the court to determine.  However, the right to park the trailer depends on that separate agreement, whatever that is, and is an additional issue for the court to determine.  If you are looking for legal representation you may cocontact me any time 24/7 365 days at 714.967.1300.