Can my parents apply to live in USA legally?

Nepal has been designated for TPS, so that may be an option (if they arrived by 6/24/15). Although the registration period expired a few weeks ago, they may be able to file late. Separately, if/when you become a US citizen, you would be able to sponsor your parents for a green card, even if they overstay their visas. You should consult with counsel regarding both of these issues.

I agree with my colleague. Contact a US immigration attorney to review all options.

TPS would have been the best option but it closed Dec 21,2015. It is possible to file late but not an easy thing to do. There is a refugee resettlement program and the US has accepted over 60,000. They should contact one of the many organizations responsible. Finally you could become a citizen and apply for them.

They may be able to apply for TPS even though the deadline has passed. If not, there may be other options available. I suggest you consult an immigration attorney for a more thorough discussion. Many of us offer free consultations via Skype or over the telephone for clients nationwide.

I agree. TPS would have been the best option in my opinion.

Short answer is no. You would do much better if you invest in a personal consultation with a reputable counsel who could explain all available legal options so you could make an educated and correct legal decision.

Once you become a US Citizen, you can petition for them. But while you’re an LPR, you can’t.