No problem off parallels, assuming the fellow off the other runway is doing the same thing - the wind will affect you both the same.
As another thicko pilot (!) I've always taken "maintain runway heading" to mean maintain the heading of the runway
and so do most of my mates, so if you guys mean something else best you tell us!! Must admit, I've never heard "climb straight ahead" - hopefully that would cause me to query what it was you wanted.