The query processor could not start the necessary thread resources for parallel query execution
There is a much better answer with more discussion and details over at dba.stackexchange.com.
https://dba.stackexchange.com/questions/47237/need-to-understand-parallel-query-execution-error
The problem indicates that your SQL server threads are already busy and unable to allocate any other threads to handle your request. You can also try to enable MARS (Multiple Active Result Sets) option in the connection string by setting it "MultipleActiveResultSets=true" if you're executing multiple batches.
This error is from building execution plan, which is trying to parallelize complex queries. It is displayed immediately, query doesn't wait for timeout. Simple queries work as a charm.
The solution was to disable Max degree of parallelism by setting it to 1. I'm not sure about performance effects of this setting, but as there are many concurrent queries on the server anyway, it shouldn't be an issue.