I live and work near downtown Dallas, and I just bought my first Airstream (72 Overlander).
There's not much, in terms of long-term RV parking, close to downtown Dallas (Turtle Creek is near downtown). As Bob mentioned in his post, Hillsboro is south of Dallas, and it can be a pretty long drive if you are commuting (between 60-90 mins depending upon traffic).
The Sandy Lake RV Park (near I-35E and TX-190 Bush Turnpike) is about 20-30 mins north of Dallas, but it's right on I-35, which is close to Turtle Creek.
Here are a couple other places around the area:
Treetops RV Village - A Carefree RV Resorts Property
Cedar Hill State Park
TPWD: Cedar Hill State Park
You may also want to read this article:
Best RV Campgrounds in the Dallas / Fort Worth, Texas Area - Associated Content
Best of luck with your endeavor!