What if block N were based not on N-1 but on both N-2 and N-3? If the bitcoin protocol were changed so that the merkle root from block N-1 were replaced in block N by the two roots of blocks N-2 and N-3, then it would become possible to mine 2 blocks at a time in parallel and increase the block frequency compared to the time each block took to propagate. Each block would still get confirmed by 6 of the following 7 blocks. The block size variance could be controlled by requiring that transactions with odd hashes go into odd-numbered blocks and vice-versa. What would be the limitations of this variant protocol compared to the existing Bitcoin protocol, assuming nobody considered one confirmation to be much better than zero?
Objections that come to mind initially, I'm sure there are more:

You don't want to split the hashing power into two halves because that would make it easier for an adversary to get 51% of that halve: 25% of the total or even less if the halves are not exactly 50-50.
The idea behind Proof of Work is to concentrate as much work as possible behind one and the same chain and make that the level-to-beat by an attacker.

If you force odd transactions into odd blocks (and vice versa) then you still have 10 minutes block times for all transactions.