So I was able to finally find the issue. I was debating if I should delete my question but maybe somebody else has a similar issue in the future.

So my services (Telemetry and Customer) are running in a Kubernetes cluster. The issue was that Telemetry's "GET" request was going through another resource. So the request that the Customer received had the Parent Spand ID from that in-between service. I reconfigured the communication to happen directly between the services and now I get the results I expected.

In the end, it was a networking miss-configuration, not an Open Telemetry miss-configuration.