if you have a dehydrator use that, if not you can air dry it in a mesh bag, or by hanging the cut stems leaves down in a well ventilated area. With my mint, I will hang a large mess bag in the garage.
@MissKathleen It was much harder on the gulf coast, but Dallas is on the edge of Texas desert. Not as hot and dry as you, but air drying will work here. As long as you have enough air circulating not to get mold, you are OK.
cut it off and dry it, leave enough to encourage new growth.
I dry mine in a dehydrator, storing it in small canning jars. I also try to use up a lot of it fresh by preparing parsley heavy recipes like green sauced shrimp and tabouli salad.
Yum, tabouli.