Although there are references to this all over the web for full framework, I had trouble finding it for .Net Core, so I thought I'd note it for others that may be searching.
On .Net Full framework, you can control the number of outgoing client connections by setting ServicePointManager.DefaultConnectionLimit to the desired number. It is 2 by default.
On .Net Core, you can set it by changing HttpClientHandler.MaxConnectionsPerServer to your desired limit. The default is int.MaxValue -- much higher than the default for the full framework.