Will an underpowered mirror host cause: The query processor could not start the necessary thread resources for parallel query execution We recently moved our 2016 sql server mirror into Azure, but it's on an underpowered machine.  Our main database server has 16 physical cores with hyper threading and 2 numa nodes.  We have set the maxdop to 16 for this machine.
Edit: I had previously set it to 8 and saw the same issues.
The mirror host has a much weaker cpu, so we had to set the maxdop to 4.
When we are running an intense query on the principle database, we sometimes get an error:
Unexpected job failure.: System.Data.SqlClient.SqlException (0x80131904): The query processor could not start the necessary thread resources for parallel query execution.

I am wondering if could be due to the underpowered mirror server running out of resources while it is trying to commit all the transactions that are being fed across to it from the principal server.
I'm not a DBA, and we currently do not have one on staff (so I guess I am the DBA), so a lot of these settings and errors are foreign to me.
Edit2:  This question is different from the one suggested because I originally followed the directions and scripts in that post to set my MAXDOP to appropriate values.
It sounds like the issue is your queries are exhausting the available threads of the new underpowered server, interestingly even with you tuning the MAXDOP setting. Please see this DBA.StackExchange answer for more details. You may need to limit your MAXDOP on certain queries further, or this could be a hint to up the server's provisioning to increase performance more.