You will need to hold an S Pass or Employment Pass for at least six months to qualify for PR application in Singapore. After obtaining the PR status, you become eligible for Singapore citizenship in the next two years. To bring your parents, you can apply for an LTVP. You need to meet the minimum salary criteria and provide some financial proof, but it is possible to bring your parents with you.
Getting a PR for person in medical profession is relatively easy. So if you are already in Singapore then you can think of applying. If you are overseas then your first task will be to get a job in Singapore. Please note Singapore does not accept qualification of all countries for medical sector.