Both words, mum and dad, have been used in English - in one form or another - since the 15th century. 'Mama/dada' is just a reduplication of a sound that babies make instinctively. In the case of 'mama', it is used in virtually all countries where the proper word for 'mother' begins with the 'm' sound, as in French, German etc.
The actual form 'dad' is far older than the form 'mum', since the former has been around for half a millennium and the latter has existed only since the early 1800s. (As 'mam', however, it's been around for as long as 'dad'.) So, it was not 'invented' but is simply a modification of the perfectly natural sounds babies make the world over.