So where is the caveat in "However, British Airways and American Airlines lounges can be accessed as part of our partner airline lounge agreement"
The way I read that the agreement provides access to the lounges. The sentence does not say "some", it does not specify that conditions may apply, it does not even specify what type of lounge (so based on that Concorde room, here I come...
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That sentence is even missing the lovely little stars which would normally indicate that there might be some conditions attached.
Finally there is no date attached to the page, so there is no indication that the answer may be old. So yes, a reasonable person could view that answer in it's own right and come to the logical conclusion that they would be permitted in a BA lounge.