If you can restrict the service to a subdomain, there are alternatives like the SAN extension that allow those third parties to avoid handling your private key at a small extra cost.

I'm having trouble understanding your comment. SANs are mandatory (current browsers don't even use CNs), how would SANs specifically prevent this? The endpoint where your terminating your traffic obviously must have the private key to decrypt it.

You need one cert (or at least only a handful of certs)—SAN entries do not need to be subdomains of the CN. Greatly reduces headache of ssl-terminating for e.g. client domains.