The Paris Agreement is either a treaty all on its own or it's a significant change to an existing treaty. Either way, the US Constitution requires ratification by the Senate.
Other countries such as Brazil had the treaty ratified by their senate.
Arguing that we now have a treaty that allows us to make other treaties without following the Constitutional process ... well.
I think the agreement itself was drafted to specifically avoid having to be considered a new treaty in a US context. So from a legal standpoint, I think it's incorrect to say "this is a treaty and must be ratified by the Senate". I will heartily agree with you that perhaps that shouldn't be the case, and also agree that we need a functional system of government where these machinations aren't necessary to pass policy, but I'm not sure there's a constitutional requirement here per se.