Does a headwind/tailwind affect your rate of climb? I am still confused whether a head wind or tailwind would affect the rate of climb. Since in a HW/TW condition our a/c is basically still climbing in the same parcel of air, it is just that parcel of air is moving with or against our course hence increasing/decreasing the horizontal distance covered to get to the target height. This means that the time needed to get to the target altitude is not affected at all, but the horizontal distance is.

Yet my friends have been insisting that with a headwind an aircraft would climb faster to the target height, as the headwind helps create more lift by increasing the amount of air passing through the wings.
A steady (continuous) head wind (or tail wind) will not affect your climb rate, only your climb angle. It means you will reach a specific altitude in the same time interval, but your ground distance will be affected.

A head wind increase (as in a gust) will momentary increase your indicated airspeed, which you can trade for a (momentary) increase of climb rate. Similarly, a tail wind gust will temporarily decrease your indicated airspeed and you might have to pitch the nose down a little, to maintain airspeed and therefore reducing climb rate. These are only transitory effects, until the plane settles back to its original trim speed.