Parking is always a nightmare in that area.Better is to choose a taxi.
If u selected from them they will arrange the delivery for you. Or else you can hire a pick up for delivery.
You may find the rug shop will deliver it to you. If you metro, which is much easier and you find a rug just pop back later and pick it up. Parking is definitely a hassle. There is paid parking at the Hyatt Regency but that is a little walk back, a taxi from the Hyatt to souk is about 8-10hds I think. Again you could park there and if you find a rug, pick it up on your way home.
I went with my dad once, he was visiting, and so it was all a bit last minute, and I found parking awful too. After deciding I was out of luck I parked across at the fruit/veg/fish market. Was a little walk back, but the best I could find at the time. Not sure if this is a good solution or not, but it worked for us on the day! Cost was the usual parking meter rates. One nice thing was that we got to check out those markets too, and I now want to go back to do a bit of shopping there some time.
We've been recently down to the Gold Souk around Deira way but parking was pretty horrible. We managed to find a parking spot but wondered what the best time was to get there? Its just that next to the gold souk are some really nice rug shops and I don't really fancy taking my chances getting a big rug back on the metro ;) I'm kind of figuring that if anyone knows it will be one of you lovely ladies