NASA SEXTANT mission demonstrated pulsar navigation to about 10km of error, and should be valid through interstellar space. This parallax method seems not really in the ballpark? Still pretty cool they are able to teach an old dog new tricks given New Horizons launched in 2006!
Once the internet started eating magazines, like many other magazines, New Scientist turned into a clickbaity, ad-heavy rag.
It's possible it's improved recently, but many people wrote it off a long time ago, which is probably why you don't hear much about it.
https://arxiv.org/abs/1711.08507
I wonder if it's really that easy to extend this to interstellar scales. If you move 300 km, a 1-ms source is reset to its original phase. Go out to New Horizons distances, and all your phases are millions of cycles offset from Earth, by highly jumbled fractional parts. Is it mathematically feasible to reconstruct everything then? Or: do you maintain a continuously-updated model across the entire mission—and hope that you don't lose count of your cycles?