In my mind a SIMD version would work by predefining a string buffer 15 elements long with Fizz in positions 0,3,6,... and Buzz in positions 0,5,10. These are constants that don't ever change. Then vectorized code would only write to positions that change, the numbers: 1,2,4,7,8,11,13,14. Most of the time these positions would have fixed width too (lengths of large numbers don't change often) so you can use constant write addresses. So 8 SIMD threads could handle a block, and write everything blockwise.

Same idea could be used to parallelize for a GPU.