Q. can they specify who I can and cannot rent two or require a background check from anyone I've rent to if I in fact already passed the background check in order to own a trailer on their lot?
Yes, if you read the laws that I cited, they state in part:
(1) A mobile home park owner may evict a mobile home owner, a mobile home tenant, a mobile home occupant, or a mobile home only on one or more of the following grounds:
(a) Nonpayment of the lot rental amount. If a mobile home owner or tenant, whichever is responsible, fails to pay the lot rental amount when due and if the default continues for 5 days after delivery of a written demand by the mobile home park owner for payment of the lot rental amount, the park owner may terminate the tenancy. However, if the mobile home owner or tenant, whichever is responsible, pays the lot rental amount due, including any late charges, court costs, and attorney’s fees, the court may, for good cause, deny the order of eviction, if such nonpayment has not occurred more than twice.
(b) Conviction of a violation of a federal or state law or local ordinance, if the violation is detrimental to the health, safety, or welfare of other residents of the mobile home park. The mobile home owner or mobile home tenant must vacate the premises within 7 days after the date the notice to vacate is delivered. This paragraph constitutes grounds to deny an initial tenancy of a purchaser of a home under paragraph (e) or to evict an unapproved occupant of a home.
(c) Violation of a park rule or regulation, the rental agreement, or this chapter.
So if the park rules require a background check on all occupants and that the occupants not have any felony convictions, then it is possible that your niece could be excluded or evicted.