I was wondering if its possible to catch something like rat lungworm or other worms and parasites via dropping a cigarette on the ground and putting it back in mouth?

If what you drop falls onto an area where there are infectious organisms then yes it is possible to conract the infection. What Mama say don't put that in your mouth dear! You are going to have to use your judgement on this one. I might add "stop smoking" it is much worse for you than the parasite.

AT 23 and smoking, you should be worried about lung and or oral cancer in the future. If you didn't smoke than you would not have any worries where you drop your smoke, but if dropped in a unclean area than you could catch all of the above as well as other diseases.