How soon can I sponsor my parents to visit Singapore? Also, can I sponsor them to stay permanently with me?

If your parents are visiting for a short time, they can apply for a tourist visa. You do not need to sponsor them. However, if they want to stay with you for a longer time, you can apply for an LTVP as soon as you become a PR. If you apply for Citizenship in the future and it gets approved, you can sponsor your parents for the Singapore PR application if they are elderly.