It depends on the parasite. Most of the nematodes / worms are reptile specific. There are a few reptile parasites like crypto and pentasomids that can cross infect mammals. (Infected mice are believed to be a vector of crypto).
The most common worms in leos (from what I've heard)are pinworms. The pinworms are usually contracted from eating crickets and are specific to reptiles. Leos infected with pinworms usually show no symptoms. To keep pinworm levels low, remove feces promptly and don't leave uneaten crickets in the tank for extended periods or they'll eat the feces and transmit more worms. Leos can also be dosed with pancur (preferably under the guidence of a vet) to eliminate the pinworms.
If your leo is sick and is diagnosed with pinworms, but doesn't respond to pancur get more fecals. Pinworms are easy to find. Far more dangerous parasites such as coccidia and cryptosporidium are not see easy to find and are, thus, often missed. Bacterial infections can also produce the same symptoms as a parasite infection.
-Alice