I get about half the mass of Venus, at 15000 a.u. to mimic the barycentre wobble caused by Jupiter. Then if we increase that mass, in theory, we could detect changes in barycentre accelerations. (It's easier to think of the barycentre moving for the super duper hard sums)
But, we cannot use parallax of nearby stars, we would have to use pulsar frequency variations. Those measurement are getting better but they're not good enough just yet, you can hide planets in the error bars.
This planet way out in the boonies, a rough Venus mass, will have a period of about 1.7 million years. This would mean that the Sun would rotate about 0.7 seconds of arc per year. Right so let's use that model again, where the sun is a billiard ball, the Earth is at 7.6 metres, Jupiter at 40 metres and Ignorethisbarrel's planet, "Pepsi" is at 144 km. (To give a better idea of the scale, the Earth has a diameter of 0.5 millimetres) Let's have this absolutely on a flat plain, with circular orbits and we want to get the parallax of a star at one parsec due north. When the Sun, Earth and Pepsi are in a straight line.
We take a picture wait six months, take another picture. We get one second of arc, and of course we can measure up to about 0.4 seconds of arc. Pepsi though has moved the whole Solar system round by half of 0.7 seconds of arc. The upshot is, that we are not taking the two photos at 180 degrees apart but slightly more. That "slightly more" is another base line, and it's too tiny to be of any use.
Righto, The italian guy's argument, is that we should compile a relativistic ephemera and then using that we can improve on those error bars for pulsars. He goes on to say that using that model there's a ninety five percent probability that there's no planet x out there.
You might think this is done already but as far as I know it's only done for Mercury. It might happen but Newton is good enough for any orbiter missions. Tiny relativistic effect we can leave to the geeks who can't dance, and think slide rules are Freudian.