I understand that I'll be taking over the leases what paperwork should I ask the current owner for at the closing?

Check your closing paperwork.

Escrow will take care of any necessary paperwork.

It's a 5 plex but city officially sees it as a 4 unit because when the original owner filed the paperwork that was the classification given.My seller tried to get a variance, but city said they wouldn't qualify.

Or can my parents just notarize the paperwork and get the building permit?

Consider it a type of transaction fee for the agent doing the paperwork and negotiation for you.

A lot of times there's all sorts of litigious talk but not that common for it to actually turn into paperwork.

The not paying 4-7% part for management is tricky....I'm still active duty....plus I would have them do the paperwork also for taxes and leasing.

Then run credit and criminal checks on the new people, and put them on the lease or evict them.Tell them you need to see registration paperwork for each vehicle parked at the property. 

Any buyer will want to see them first.2) Make sure you have ALL the tenant paperwork like leases, income verifications and re-ups AND THEY ARE ALL CURRENT3) Make sure the rents are calculated right based on incomes and FMRs and reflect any UAs4) Understand the financing and any payoff provisions.