On the link quoted in the question it does not appear that the President is indeed at an airport. There is a large tree adjacent to him, and park benches. Thus the discussion about whether airports have one or more tarmacs doesn't seem to apply here.
Mr Obama and his daughter Sasha, 14, walk across a tarmac in New York on Friday.
For me that sounds awkward. How many tarmacs are there in the vicinity?
He also doesn't appear to be walking across it, but rather along it. I would have written:
Mr Obama and his daughter Sasha, 14, walk along a footpath in New York on Friday.
(substitute "sidewalk" for "footpath" if you want)
Later in the article the author says:
Afterwards my colleagues and I head across the tarmac to an Osprey aircraft, propellers spinning.
Just as a guess, this is a simple typo by the journalist. At one point it is "a" tarmac, later in the same article it is "the" tarmac (although admittedly this time at an airport).
(Edited to add)
Now that Peter Shor has cleared up that the first photo does not relate to the sentence underneath it, I am going to vote for the "the tarmac". In much the same way you might say you saw something "lying on the ground" (rather than "lying on a ground") I suggest "the tarmac" is better.
You could qualify it (eg, "a football ground") in which case "a" is better. In the same way you could say "an airport tarmac" (as opposed to, say, a school ground tarmac).
INCORRECTLY to describe airport aprons, "ramps", and runways.– Mazura Jul 23 '15 at 21:55