So I have been reading a few things about fourth era lore on UESP since I am playing skyrim again. There is an event called "the accession war" and argonians apparently invaded morrowind but were stopped from advancing by the redoran. So do Argonians rule southern Morrowind?

Yes, though we don't know how far north they got. I doubt they got beyond the southern marshes on the Black Marsh-Morrowind border, I just don't see Argonians as being able to hold the ashlands for any long amount of time. I supect that, since they have ruled it for almost two hundred years by now, the lands they annexed are probably not considered Morrowind anymore, but a part of Black Marsh, especially since the line between the two is a bit blurry in that area.

It seems most likely that they only occupied the parts of southern Morrowind that they felt was theirs. These include territories lost to Morrowind during the Arnesian War, as well as older territories they laid claim to, so thought would make good buffer territory, perhaps even including the city of Tear (formerly a great slave trading center). I just can't see the Argonians occupying anything but swampland though.

However, the border remains completely unstable and Argonian raids, something unthinkable under the Tribunal, seem to have been a common occurrence throughout the Fourth Era. We have reports of them raiding as far as Solstheim, which suggests that, while the Redoran may have turned back the original invasion, the security situation in Morrowind continues to remain in complete tatters.