Your connections are waiting for previous connections to close, thus the timeout. The default limit for concurrent connections to the same host is 2. Try increasing that limit before entering your Parallel
call:
System.Net.ServicePointManager.DefaultConnectionLimit = int.MaxValue;
Read more about the DefaultConnectionLimit
here.
Property Value
Type: System.Int32
The maximum number of concurrent connections allowed by a ServicePoint object. The default value is 2.