By your logic I could never feel proud of my parents, and that would be a real shame.

You've followed NONE of my logic. I said it was OK to be proud of achievements.
For the same reason I can be proud my Grandfather helped liberate Europe in 1944-45. For the same reason I can feel proud that by Grandmother and Great Grandmother resigned from the Daughters of the Revolution when they refused to allow Marion Anderson to sing. It feels good to be proud of those we love and with whom we have a connection, even those remotely via ancestry. There's nothing wrong with it.
EXACTLY! You are proud of their achievements, not that they are who they are. You aren't proud of your family name, but of what your family did.
THAT is my logic.
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