After parking ends each time, do I need to save a new backup of wallet.dat in order to make sure the backup is updated with the new total balance from parking ends?
You can if you want to make sure, but it is not necessarily, because the wallet always synchronizes with the network.
Make sure you keep wallet.dat file in a safe place, like flash drive.
But if the wallet.dat file is stored in a flash drive not connected to the internet, how can it synchronize with the network to update the new amount of coins?
A wallet.dat stores everything needed to access your accounts, it doesn't store the balance : That's what the blockchain is used for.
Consider it as a key to access your safe, nothing else.
The only true backup you need is a copy of the private key(s) for your wallet address(es). Having the wallet.dat backed up is still a good routine, but if it gets corrupted or deleted and you don't have your private keys, then you're screwed.
Keep a txt file and/or hand written set of wallet address / private key pairings; so long as you have the privkey, you can recover your wallet from any MNX wallet software client (which will add it to the current wallet.dat in use). You can recover it to multiple wallet.dat files (if you were to have, say, a master wallet of all addresses you use, as well as wallet files with individual addresses).
Regardless of any of that, if you're going to back up your wallet.dat, then keep a string of backups. They don't all have to be sync'd regularly; just regularly toss the .dat into a backup folder and 'Copy, but keep both files'. This is in case of the previously mentioned file corruption.
I personally keep a txt record of address/privkey pairings on a thumb drive alongside my various wallet files. And a copy on my spare encrypted & locked HDD. I just need to print/write the address:privkeys out to complete the trifecta. :p
But my question is how can the balance get updated on the stored wallet.dat file when parking ends each week? If it's just on a flash drive, it would need to be updated every week with new dat file, correct?
The balance is kept on the blockchain, not the wallet. You don't need your wallet.dat to access the wallet address and its balance on the MinexBank page. You only need access to your wallet.dat via wallet software to send your coins to the exchange (or another wallet). If you wanted to be diligent, you could just backup your wallet.dat after every parking by tossing it onto the same thumb drive (Copy, but keep both files); again, in case of file corruption.