iControl REST - Finding a pool with a specific member

As Dario mentioned, use ?expandSubcollections=true to include the actual information (instead of just links) on the pool members. To extract the pool names that contain a certain pool member, I would use jq.

Example: Extracting the pool name(s) that contain the member "CentOS-internal20:443"

$ curl -sku $PASS https://$HOST/mgmt/tm/ltm/pool?expandSubcollections=true | jq '.items[] | select(.membersReference.items[].name == "CentOS-internal20:443") | .name'
"Pool-CentOS443"

Be careful, though. Since your BIG-IP has 500+ pools, the response body is expected to be fairly large. Due to its size, the call may pressure the Control Plane and mcpd (configuration management daemon), and subsequently the call may time out. Test it while the system is not busy.

For more information on the iControl REST options, refer to iControl R EST User Guide (PDF) in iControl REST Home.