Short answer - not really, parallax for the closest stars is right on the limits of resolution for a good ground-based amateur equipment.
The nearest star would show a parallax angle of under an arcsecond. (The Parsec even takes its definition based on one arcsecond of parallax.)
Ground based amateur observations are probably limited to an arcsecond at most, so it's probably not possible to measure them.
A special (and totally fictional) case - if a nearby star like Alpha Centauri were exactly between us and a much more distant star, and it were possible to see them both without Centauri overpowering that distant star, then maybe it would be possible to observe them as a close double. That's not true in practice however.
Update:
Prompted by Rob's much better answer, the earliest parallax determination I've been able to find was using
the 6.2-inch (157.5 mm) aperture Fraunhofer heliometer at Königsberg
(Source: Heliometer article on Wikipedia.)
In purely aperture terms, that's an amateur-sized instrument (with some admittedly very precise measuring gear built in.) So my suggestion now is to ignore my initially doubtful response and try measurements, perhaps using a camera at fairly high magnification as Rob suggests...
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