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 Earths 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.
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