A parent or legal guardian can share their Bnet account with one minor they are legally in charge of.
Only you are allowed to access an account registered in your name. We don’t recognize the transfer of accounts between individuals. Activities performed on your account are your liability.
You may not share your account or password with anyone, except if you are a parent or guardian, in which case you may permit one minor child to use your account. You may not use your account at the same time, and you are liable for activities conducted by the minor child.
You are responsible for maintaining the confidentiality and the use of your user name and password. You are responsible for all uses of your information, even if you don’t authorize them. Security of your account is your responsibility.
The Desktop App (Launcher) will only allow one person to use it at a time though. So you would have to launch WoW from the folder and .exe instead then manually log in. Of course, the Launcher is required in order to update WoW so if your game is not up to date it won’t launch. (This is subject to change over time as Blizz changes game tech/launch options)
Word of warning though - allowing anyone else on your account opens you up to security issues and disciplinary issues. A silence in ANY Blizzard game is now Bnet account wide so you may find yourself unable to talk in WoW due to actions by your son.
The best advice is usually to put the minor on a separate Bnet account in the parent’s name with Parental Controls set up (controls online purchases, play time, produces play reports, controls chat functions, etc.). That also ensures that when your son reaches 18 and has to have the account put in his legal name, he gets to keep all the things he achieved that are tied to his Bnet account - which is nearly everything.
Here is a reddit thread from a while ago that seems relevant to your issue. https://www.reddit.com/r/Overwatch/comments/4f55mf/can_i_play_wow_while_my_son_plays_overwatch/
Here are some of the best replies as far as I can tell.
You might be able to get away with logging into the Wow64.exe with your battle.net account instead of the client and then letting your son log into the Battle.net client.
To clarify, yes, you can, but you can’t both log into the battle.net launcher. On each computer open the game from it’s folder instead. It’ll bring you to the game’s log in screen and you can log into both games at once then. I often play hearthstone (which does actually need to use the launcher) and WoW or Heroes of the Storm at the same time. It’s great for waiting on queues.
Again though, it is honestly best if he is on his own Bnet account even if it is in your name and you set up Parental Controls.