Will an underpowered mirror host cause: The query processor could not start the necessary thread resources for parallel query execution
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.