Click the link for Dallas city code.
Parking is covered under Volume 1, Chapter 28, Article XI.
You can park your trailer on the street, same rules apply as a single unit motor vehicle. Parked facing the right way, don't block driveways or hydrants, don't park on a lawn, wheels within 18 inches of the curb, 24 hour parking rule, etc.
Cars parked on Dallas streets get nailed on a pretty regular basis, I wouldn't park a boat on the street unless I had comprehensive coverage that extended to the boat and trailer. A lot of people wake up every morning and find that someone has side swiped or hit their car and not left any information, most insurance companies won't cover that under an uninsured/underinsured motorist policy. The coverage is probably a good idea anyway.