Please could someone explain how he manages to get a single parabola from the pairing term? All I understand about the pairing term is that a_{p}<0 for Z,N even, even. a_{p}=0 for A odd and a_{p}>0 for Z,N odd, odd.

Simple:
If A is odd, then N+Z is odd...
If N+Z is odd then one of them has to be even and the other odd (summing two odds or two evens will give an even number).
So a_P becomes zero and that term misses from the energy...
If A is even then a_P is non-zero... at this case you can have two different Z,N that give the same A which can achieve the least energy...

Thanks thats really helpful and clears up the a_p values. I might be being realy stupid here but I'll ask anyway, if the a_p term is zero then how is it a parabola? The only thing I can think of is that if the term is missing and you put N=A-Z into the equation you get some sort of quadratic in Z, which would be a parabola?