Can ASIC mining machines mine in parallel?
Yes.
Can they pool their hashing power to mine in parallel and avoid doing the same work twice?
Yes.
Are machines in standard mining pools actually competing on one block?
No. They are each working on their own separate block for every hash they compute. However, they are typically all working on creating a block for the same block height.
Do they mine in series?
No.
Or do they mine in parallel, genuinely adhering to a hierarchical tandem cooperative relationship so they don’t do double work?
Correct.
I mean, if I take 3 ASICs and stick them together, they’re just 3 independent machines, so yes I have 3x the chances of mining a block.
Correct. If they are identical ASIC with the same hashpower, running at the same clockspeed.
But if they were to share the same mining algorithm, mining in parallel and sharing each other’s task- each mining a different part of the block so to speak, to avoid repeating each other’s calculations- they would be exponentially more effective. Is this already done?
Yes.
What is the software for this?
Any standard Bitcoin mining pool software.
Are mining pools typically series or parallel?
They all work on the current block height in parallel. Block heights are handled in series (block height 1 MUST exist before ANYBODY can start on block height 2)
Is this a common configuration?
Yes.
It seems to me that you effectively asked the same question 10 times? Why not just ask it once and then wait for the answer?