I don't think there are any weather requirements.
Doc 4444, 6.7.2.2
REQUIREMENTS AND PROCEDURES FOR INDEPENDENT PARALLEL DEPARTURES
Independent IFR departures may be conducted from parallel runways provided:
a) the runway centre lines are spaced by the distance specified in Annex 14, Volume I;
b) the departure tracks diverge by at least 15 degrees immediately after take-off;
c) suitable surveillance radar capable of identification of the aircraft within 2 km (1.0 NM) from the end of the
runway is available; and
d) ATS operational procedures ensure that the required track divergence is achieved.