A properly igniting natural gas stove or oven should leak minimal gas into the air and any odor should be very transient. I've had nothing but natural gas cook-tops since I've had birds and have never had any problems, but the appliances were relatively new and well maintained. And the birds are not in the kitchen when I am cooking.
If you consistently smell gas and the stove is hard to light, you may want to have it serviced. If it is a sealed-burner kind where you can clean the components, that alone may get it lighting more efficiently. In fact, if the burner covers are not seated properly, it may delay ignition. The gas needs to ignite right away or you will smell a little bit. A very small amount is normal and should not travel very far away from the range.
If you smell gas all the time, something is wrong and you need to have it fixed asap as everyone who said that situation poses a risk to your birds first but you as well is correct.
Fortunately the chemical they put in the gas to make you able to smell it (it is inherently odorless) is very strong so a tiny leak is detectable. The odor itself is not harmful in this case, it's the gas that poses the main risk.
A good vent fan in the kitchen goes a long way, too.