What is the probability of an offspring sharing identical HLA typing as one of their parents?
If you follow your math, you will definitely not find a 1/1,000,000 chance of two random people being an HLA match, so it seems clear that the independence and equal weighting of types is not a great assumption.

We can make a different set of back-of-envelope assumptions, and just say that the 1/1000000 number comes from having 6 combined matches. In that case, you would interpret 1/1000000 as due to 6 independent matchings of probability 1/10. In that case, with a parent you only have to randomly match the other three. 1/10^3 = 1/1000, much closer to the 1/200. Further non-independence, unequal frequencies, and homozygosity makes 1/200 seem reasonable in a whole population (though individuals with more rare HLA would be less likely to match a parent or a random individual), and I presume the 1/200 number comes from actual observed frequencies rather than theoretical calculations.

Perhaps someone with more HLA expertise (I have basically none) will provide a better answer based on more specifics, but I just meant this as a quick napkin calculation for whether the numbers given are plausible, and I think that they are.