Isn't it possible a very faint star or brown dwarf could be close to the sun and exhibit very little proper motion, especially it were a binary companion to the sun at the far end of a highly elliptical orbit? Then again, it could be a straggler coincidentally traveling parallel to the sun at roughly the same galactic orbital velocity, or almost radially with respect to it. If it was of magnitude<11 Tycos/Hipparrcos would have not picked up a parallax or a proper motion at all.

It would still be visible to a moderately large scope, but it would be just one out of many in the field, there would be no way to get a parallax unless you could identify it ahead of time as a target worth studying. Most stars who are measured for parallax give themselves away by having large proper motions, which makes them worthwhile subjects for astrometry.

A specialized survey looking for an spectral infrared excess might be able to find it, though.