I agree with your parrot, the more seeds the better.
In New Zealand we have a cucumber called the port Albert cucumber, its big, easy to grow, and has tons of seeds. Unfortunately I couldn't find it for sale outside of NZ (maybe under a different name?).
A slightly smaller international alternative, still with lots of big seeds (tho not quite as many), is the Crystal apple cucumber. Its also easy to grow.
We've grown both of them and will grow both again. They get more/bigger seeds when left to mature, but go sour if left for to long. I still love them when they are sour (your parrot might to) but everyone else thinks I'm crazy.