I am a TA and had the daughter of a rock star in my class last year and will have his younger daughter next year. After being a massive fan of him and the band for years, and having seen them live many times, I found it very difficult at drop off times! I have found myself saying rather ridiculous things about the weather!!! I had the dilemma of going to see them in concert, as I would have done if he wasn't a parent at work, or not bothering. Of course I went and had an even better time!!
As for getting past the obvious blushing - on my part - and general feeling of awkwardness, I don't think that goes away if you admired that person before. I know members of staff who have developed a "taste" for their music only since he joined the school which bothers me a lot! Lol
Just treat this parent the same way you treat the other children's parents. Many years ago I taught the son of member of a well known (and very "way out" pop group - weird in fact). When he came to school to drop off/collect his son he looked perfectly normal but looked very weird on stage! I was not at all starstruck as I was not a fan of his particular brand of music so found it easy to treat him the same as other parents. I was a bit worried about what he might say at Parents Evening as he was a straight talker but it went well as his son was doing well. He complimented me on the class assembly we had done earlier in the week. He said it was the best assembly he had seen at the school in the 3 years his son had been there. I must say I was very proud of that and remember this parent - and his son who was a lovely boy - very fondly.
fwiw, I meet a number of stupidly famous people through a relative of mine, and tbh I do get a bit giddy inside, but, I can imagine how hard it must be living your life with everyone staring at you so most of the time I am careful to be over chilled.