Please don't get a dog if you're not allowed.
Put yourself in the owners shoes- would you like someone to lie to you and go against your wishes of what goes on in your house? Your house that you've spent hundreds of thousands of dollars on, paying for repairs and maintenance to keep it looking in great condition?
If you have your pet indoors, it's not as simple as having carpets cleaned. Pets have a smell that is difficult to remove; people that own pets probably don't notice, but they do. Cats and dogs also shed fur onto the carpets, the fur gets caught up in the carpet and you cannot get it out. What if the owners have allergies and are hoping on moving back in there one day? If the pet soils the carpet, it'll sometimes go through all layers, you can't get that out by steam cleaning.
Outdoors, a dog may ruin the grass. I know my dog (at my mums house, because I'm renting), has her favourite tracks to run and so the grass has died down in these spots, and teh dirt has compacted so it's very difficult to get it to grow back. What if your dog does this? It's not like they can fix grass in the few days between tenants. Also gardens.
Sorry but I really think it's terribly dishonest and incredibly selfish.
In addition to this, a dog lives for what, 15 years? Do you think you will have success in either having acceptance of a dog or 'hiding' a dog for the next 15 years? What happens if you get found out? You will either have to pack up and find somewhere else to live quick smart, with a blemish on your rental record, or you will have to surrender the dog, which is unfair to the dog, and also to you and your family who have grown to love it.
It's all a very bad idea.
That said.... I have rats, without asking permission. However they are in a cage in the laundry, so they are never near carpet or any other surface that could be damaged. If the kids want to play with them they do so in the bathroom, again, so there's no damaged surfaces.
I'm also getting a scorpion (he's coming in the mail in the next few days!) but I don't think I need to ask permission for him anyway because I think I'm allowed something in a tank. He's not exactly a cute and cuddly pet though. But as PP's have mentioned, if you rent, you really need to accept that you are unlikely to have pets.