Looking at "mata-pita" (mother-father), Karan thinks it means "a father like a mother" — one word…

Why

Karan is applying a very common compound-word pattern — two words joined so that one describes the other, as in 'नीलकमल' where 'नीला' describes 'कमल'. But compound words (समास) come in several kinds, and 'माता-पिता' is a different kind — both words here are EQUAL, neither describing the other; instead they show two separate, equal things joined by 'and' — 'माता और पिता' (mother and father). This kind of compound is called द्वंद्व समास (dvandva samas), where both parts are equal partners. Test it: 'भाई-बहन', 'सुख-दुख', 'रात-दिन' — in all of these, both words are equal, joined by 'and', not one describing the other. His sister is right: 'माता-पिता' means 'माता और पिता' — both words are equal here; this is द्वंद्व समास (dvandva samas).