Can I get notifications for specific emails sent to my Gmail account on iOS?
Either set up your Gmail account using Mail or try the Google's official Gmail client. Both use iOS's built in notification system. SMS and iMessage are entirely different protocols and do not support IMAP or POP3 (the protocols used for Mail).

As for parsing the notifications for only certain emails, I'm not sure that's at all possible if you'd still like to receive notifications for other emails too.