I get asked this question via e-mail and on the phone all the time... please don't call!
Many of the old Atlas parts were sold to Clausing back when Atlas went out of business. It has been MANY years so it is hard to say just what parts they may still have and I know the prices tend to be high but it is worth a shot if you really need something.
Sears also has been reported as having some spare parts. With that said, I have not heard of anybody being able to get replacement parts for an old Atlas lathe in years so that is likely more of a long shot then Clausing.
The best place to look for parts (that I am aware of) is EBAY. I have a series of searches programmed into Lathetalk.com that I use to keep an eye on parts and tools that I am trying to find... I find it easier for me to click a page and see what is ending over the next day or two...
Lathetalk EBAY Searches
That page needs to be redone now that the server move is over but the page does work as it is.
Good luck and you may discover that your lathe is worth more if you tear it down and sell the parts out of it then the lathe is worth as a whole... This is because other people are also looking for these old parts. I was able to get all the parts I needed to get my old lathe running off EBAY but it did take some time.