Do you need both client and server side support for TLS 1.3 record padding? When you enable record padding for a client, does a server need a specific option to support this? Or does the other end only need to support TLS 1.3 as well and can handle the padding automatically?
When you enable record padding for a client, does a server need a specific option to support this?

It shouldn't; record padding is a part of the RFC which is not listed as optional; I don't see anyone could claim to support TLS 1.3 without understanding record padding on decryption.