I have checked the vents in the house and my sewer lines keep filling up.

Sewer lines are usually the responsibility of local government.

Except from the street to the residence... which is usually the responsibility of the tenant or property owner depending on contractual agreements. From the street to the treatment plant its the authority who runs it.

But the trailer park owner would be responsible for his sewer lines underground inside the park, right?

Depends on what cause of the problem is... if someone has been flushing things that have no business being flushed (feminine napkins, tampons, grease etc)... and it caused a clog.. that would be on you. Tree roots, collapsed pipes... etc would be the trailer park owner.