Security issues with computing parallel hashes?
Are there security problems with such a design that wouldn't exist with a normal hash algorithm?


At least not for collision resistance.

Suppose you could come up with a collision for the composite construction $H_p$ based on the hash function $H$. Then you'd have two inputs $x\neq x'$ such that $H_p(x)=H_p(x')$. Then either there is a collision on the final hash invocation and we can reconstruct the relevant inputs following the description of $H_p$. Alternatively all the inputs to the final invocation are equal between $x,x'$ and then the slices that differ produced a collision in $H$ which we can reconstruct.

So every collision for $H_p$ implies a collision for $H$, but because we assume $H$ to be collision-resistant there is no efficiently findable collision for $H_p$.