I am a paediatrician. I did this at the beginning of my career alot. This was not about pulling rank or being slapdash. Firstly, it was what we did. All the doctors, nurses etc. Everyone learns by example and so you do it too. Secondly as a children's doctor often the parents names aren't easily accessible in the notes, and Mr/Mrs child's surname isn't always right either. Thirdly, in my head I thought it was kind of nice, because I thought I was stressing that I knew they were important.
However, in a similar thread a few years ago many people heartily disagreed with the above points. 😁. With a bit more thought, I decided to stop doing it. It is an extremely ingrained habit and I lapsed alot at first and still do occasionally now. But now I introduce myself at the beginning of a consultation and ask for the parents names. I then use them throughout. When I write a letter, I write it as Jane (Emma's mum) the first time I refer to the mum and thereafter just "Jane". When I bring other doctors in I also introduce them, for example "Tom and Sue, this is Dr Smith, Dr Smith, this is Tom and sue, Fred's parents"
In a neonatal unit I worked in I changed the cot cards so that the mum and dad's name were on them. This made it easier to talk to the mum and dad at the cotside, as while I can remember a name for a consultation, I struggle to remember a whole unit of parents!
What I would say, is I am one of the only hcps I know that do this. Parents often look quite surprised when I ask their name.