You can also try linking their paper applications to the online accounts so that you don't miss out on any update and/or if IRCC requests for any additional documents.
Background checks: IRCC is doing background checks (security/criminal/fingerprinting, etc.... maybe some of these or maybe all of these and some more) on your parents' applications. There's no time frame within which these checks will be completed. And, in fact, there's nothing you can do to expedite a decision until the reports are received by IRCC from external agencies.
In short:
You'll need to wait patiently because sometimes background checks can be completed in a few weeks and in other cases, it could take several months. Until then hope for the best and don't lose heart because of the delay