Answer: Based on your description it could be the unvariegated form of Aegopodium podagraria, although I am not aware of people suffering a skin reaction to that plant.
For reference, here is a link to a photo of poison ivy in bloom (scroll down on the left, it is the best photo I could find in a quick web search). The flowers are small and yellowish and not at all as showy as Queen Anne's Lace.
For a more positive identification of the plant, I would suggest you consult with your county extension. Based on that, they should also be able to tell you the best way to eradicate it.
It is also possible that the skin reaction came from somehere else entirely, somehwere that the dog had been and then you came into contact with it on the dog. I'm sorry I can't be more helpful.
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