The idea:

We split the input in maximum of 16 groups (32 bits each). So, with every parrot you transfer a group number (4 bits) and some number (4 bits - from 0 to 15). Now, we can pass the numbers and distinct the initial value of group by different combinations of counts of numbers from 0 to 15. How to count it? If we pass X numbers, then the amount of combunations is C(15+X, 15).

So, when the sum C(15,15) + C(16,15) + C(17,15) + ... is >= 2^32?

Counting until 34, the result is 4059928950, but until 35 - 7307872110, which is just enough. So on every group maximum of 20 numbers should be passed.

So, on every 4 bytes we need one group, so the ratio is: 20 * |` N / 4 `|. Which gets us ratio of 5. The problem is, if N is not divisible by 4 - we still need the same amount of groups, so the ratio gets slightly bigger than 5. However, if N is 63, not 64, the last group is not in range [0..2^32), but in range [0..2^24). If we assign our combinations cleverly, so that smaller number gets smaller amount of numbers to pass, the ratio of 5 is still present:
C(15,15) + ... + C(30,15) > 2^24
C(15,15) + ... + C(25,15) > 2^16
C(15,15) + ... + C(20,15) > 2^8

Now we need to get a bijection between combinations and actual numbers, but it can be done with DP, and the big number aritmetics is not needed here.

I can provide a source code, which I coded after IOI (on IOI I coded similar idea, only 64 groups (8 bits each) - that approach gives 98 points with a ratio of 7), if I can find that source code on one of the flash drives.