  We present a systematic search for parallax microlensing events among a total
of 512 microlensing candidates in the OGLE II database for the 1997-1999
seasons. We fit each microlensing candidate with both the standard microlensing
model and also a parallax model that accounts for the Earth's motion around the
Sun. We then search for the parallax signature by comparing the chi^2 of the
standard and parallax models. For the events which show a significant
improvement, we further use the `duration' of the event and the signal-to-noise
ratio as criteria to separate true parallax events from other noisy
microlensing events. We have discovered one convincing new candidate,
sc33_4505, and seven other marginal cases. The convincing candidate (sc33_4505)
is caused by a slow-moving, and likely low-mass, object, similar to other known
parallax events. We found that irregular sampling and gaps between observing
seasons hamper the recovery of parallax events. We have also searched for
long-duration events that do not show parallax signatures. The lack of parallax
effects in a microlensing event puts a lower-limit on the Einstein radius
projected onto the observer plane, which in turn imposes a lower limit on the
lens mass divided by the relative lens-source parallax. Most of the constraints
are however quite weak.
