I'm out of my depth also, but my rambling would be similar to HelloU.
Firstly I don't know about in Sweden, but in Australia a change of ownership, triggers a capital gain event.
Secondly I doubt Comsec would deal with shares on the Swedish stock exchange, you would have to check with them.
What I would guess would be, he would have to transfer them into your partners name in Sweden, on their share register. Then she would have to find if any of the registers here deal with the Swedish share register, if not she would probably have to sell them over there.
It may end up easier for her father just to sell them, but it will be an interesting project for you to work it all out. IMO