I too am having issues with PS4 party chat and receiving NAT errors. I have a TalkTalk super router – HG635. This started happening around 3 weeks ago.
I have tried with a LAN cable and also wireless, gave the PS4 a static IP address (outside of DHCP range), put the PS4 into the DMZ, forwarded the relevant ports to the console, turned off the router firewall, factory reset the router but I still get party chat NAT issues. The error happens every time with party chat, it’s not intermittent.
I have not yet tried upgrading the router firmware if that’s even applicable or trying another router. This may just be TalkTalk to TalkTalk but I cannot check this.
Does anyone have any other suggestions that I could try?