Is there an English idiom for trying to do two things at the same time and failing at both of them due to splitting your effort?

Chasing two rabbits from the widely claimed proverb:

"He who chases two rabbits will catch neither."

Betwixt and between

Torn between the two tasks

Stuck in the middle

Double minded from the New Testament:

For he that wavereth is like a wave of the sea driven with the wind and tossed. For let not that man think that he shall receive any thing of the Lord. A double minded man is unstable in all his ways.

Moving farther afield:

Between a Rock and a Hard Place

By implication, Hemming and hawing

Between the Devil and me

I think you should go with "spread yourself too thin"

spread yourself too thin: to try to do too many things at the same time, so that you cannot give enough time or attention to any of them. "I realized I'd been spreading myself too thin so I resigned as secretary of the golf club"

(Definition of ‘spread yourself too thin’ from the Cambridge Advanced Learners Dictionary & Thesaurus © Cambridge University Press)

Or, as Bilbo painfully explains to Gandalf, "Why, I feel all thin, sort of stretched, if you know what I mean: like butter that has been scraped over too much bread."

Falling between two stools is another that works quite well. It meets all three of your bullet points:

• attempting two things (or multiple iterations of one thing)

Multiple iterations of sitting on a stool, check.

• both of which you might succeed at, if you addressed them serially

Sitting on a stool isn't hard, check.

• but both of which you fail at because you address them in parallel

Precisely what happens: you try to put one butt-cheek on each stool, but end up sitting on the ground between two toppled stools.

It's a bit crude, but You can't ride two horses with one ass (also here) or one of its variants seems to be getting at exactly your meaning.

"Multitasking to a standstill".

It's more often used of computer multitasking, when the overhead involved in making the multitasking happen and ensuring it doesn't cause bugs when two processes want the same resource make things slower than if you hadn't used multitasking.

But I've found it can sometimes describe human activity very well, too.