perhaps if you are from some distant backward province it is different.
I must be from a distant backward province then, because I wouldn't consider the word to be offensive. I'd say that it is old fashioned, archaic even, and it's not a word that I'd use myself (I'd probably go with 'mixed race') but not offensive.
The Jim Crow Museum of racist memorabilia apparently has no problem using the word
ferris.edu/news/jimcrow/mulatto
the PC crowd will have you believe the entomology of mulatto has its origins from mulas in Spanish
Indeed, such Spanish people would have just called them 'mulas', in the same way that they used 'marrano' ( pig in Spanish) to describe Jews that converted to Christianity.
Of course, with that said, there is a distinct chance that 'mulatto' might go the same way as 'chinaman' (i.e. wasn't an offensive term until some people claimed it racist and then became impossible to use because some people considered it to be racist).