a girl can also say мой друг in the same meaning (e.g. мой парень sounds awkward when a man and a woman both about 50 years of age are dating, because парень is a young guy)...
моя подруга would be used more often for 'my girlfriend', моя подружка is a diminutive form...
we don't seem to have a universal unisex term, партнёры might fit, but this sounds quite formal, that's why people rarely use it in casual conversation...
so the neutral terms are друг (парень if younger than 40, perhaps) and подруга.