I personally did not experience this in a way that related to longing for a parent, but isn't it also how people very often fall in love with a new partner, or prospective partner? A strong sense that there is something almost mystical and magical in the encounter and in that person. Finding a soul mate as they say. My version, when I still sometimes tend to feel a mild form of that sort of excitement and kinda child-like longing and enthusiasm, is when I meet someone new that I sense (or imagine) highly similar to myself and we have great sync, think alike, do things in similar ways. I have experienced this with people I don't know how many times since my early childhood and it has lost from the intensity a great deal with so many repetitions, but just never seems to get old completely.

When thinking about something that feels more like parent-child, what comes to me easily is my mentor-mentee experiences. As I got older, I've learned that I can also experience it from the opposite end, with someone that I am mentoring, when it feels and functions like a natural great fit. Just had one of them a few weeks ago with a student. After a short while, that sense of similarity hit hard both of us and working together became quite intoxicating (in a good way). Also very efficient. I tried to remain as professional as possible and would not respond to his more personal reactions, but I saw in him many instances that felt like that metaphor in the OP, he also sometimes could not control it and overreacted a bit publicly. But it was not merely in his imagination, it was a response to something very real and mutual. I was actually a bit uneasy with myself when he left (was an intern), how much I missed him and kinda craved his presence for a couple weeks, almost like craving a drug. I have not experienced similar from the other end for a good while, in relation to a mentor of mine (or a therapist), but had many many times when I was young. I also got into quite a few "inappropriate" relationships driven by that instinct, desire, love - call it whatever. Often got very good and meaningful experiences out of it but not only that, sometimes it can be very painful with the limitations of reality and morality. My experiences never truly feel hierarchical or parent-child-like though, for me it does not have feelings of loss or longing for something I've never had, much more like equals even when there are large age or other kinds of gaps. Kohut's Twinship concept is what describes it best for me, it is something I intensely relate to and am prone to. I've found and experienced it so many times in so many forms and still works. I guess it is because it's just the kind of interpersonal experience I like the best, never gets enough I would say most of them fulfills the desire in their own way but the limits of reality or human decency also usually restricts them - perhaps that's why it comes back again and again.