Does doing Help / Update from within Elements Editor get you ACR 8.7 or newer? According to the Adobe ACR plug-in version Apps list PSE13 should work with ACR 8.7.1 which would be new enough:
My Elements has ACR 9.0 even though 9.1 was installed for Photoshop, so there doesn’t seem to be an updater for ACR 9.1 that also installs it for Elements, and it’s possible that Elements won’t recognize ACR 9.1 at all.
If necessary, here is a ProDesignTools page with links to the ACR 8.7.1 installer.
It is for CC but may work for Elements. If not, you can always unzip the setup.zip and go looking in the folders for a large file named 1002 or 103 or similar, that will be the camera raw.8bi file that you can just rename, and then copy to the folder on your hard-drive that has Element’s Camera Raw.8bi. I’d rename Elements’ one to something else without an .8bi extension, first, in case things don’t work and you want to go back.