hi my co-worker had the exact same issue and same age as your cat, we looked at her cats teeth and it was covered in tartar, she took the cat to the vet and got her cat's teeth cleaned and now he's eating like crazy and has gained his weight back, hope this helps...
It's not unusual for an animal, ANY animal, to lose weight as it reaches its geriatric years. It's pretty common, actually, and it can be caused by any number of things. Thirteen is getting up there, but it's not terribly old for a cat any more. It's also very normal, though, for a cat his age to slow down and not play as much, in fact, they start slowing down well before 13 in many cases.
As long as he is holding down the food he is eating and isn't having any problems with diarrhea, chances are the changes you are seeing are just a part of the normal aging process. It's only natural that you are worried about him, though, because you love him, and when you love something, you worry. If it would make you feel better, ask your mom if you can take him to the vet for a checkup. Maybe have him do a fecal test to see if he has internal parasites. That could account for his weight loss. But I think that you'll probably find that he's just a normal, 13 year old cat.
I have one that's the same age, and he has the same symptoms that yours has. He's not as thin as you have described your cat to be, you can't feel his backbone, but he has slowed down a LOT in the last four or five years and he even LOOKS like an older cat now. But he is healthy and I look forward to having quite a few more years with him.
Just keep an eye on him. If you see him developing symptoms of illness, take him to the vet. Other than that, just enjoy having him around. It's good that you are worried about him, but don't worry about him so much that you forget to enjoy him. I think he'll probably be around for a few more years yet, from the sounds of it.
Ghilly