In parallel circuits, branch Amps is inversely proportional to branch Ohms but what determines which branch a particular electron takes.?

At the electron level, it's practically impossible for all the branches to have the same resistance. The electron reacts to the LOCAL electric field, mainly from the eight or so closest atoms and the same number of free electrons. These atoms and electrons are constantly jiggling around. In the very rare cases when an electron feels the same field in two directions, it just randomly picks one or the other.

Plus, it's interesting that the electrons move really slowly, at a typical speed they take 20 minutes to move one meter. The electric field moves at the speed of light; in copper that's about 95% of the speed in vacuum. So the field moves over a billion times faster than the individual electrons.

There are many possible similar questions like where does a particular grain of sand go when you pour a pail full on the floor, where does a molecule of water go in the ocean, a molecule of air in the atmosphere. The answer is that we might call it random because the minute details that determine these things are far to complex and irreverent to worry about. But the probability is well known and the overall distribution of electrons, grains of sand, molecules or air or water etc will be predictable. The universe operates by the laws of statistics.

How would you identify the electron? They aren't numbered. Trying to keep track of them would be futile so it doesn't really matter which electron goes where in a circuit.

I don’t think your question can be answered with certainty as the identity of a specific electron is really tough to find. They kind of look and behave alike. None of the electronics I’ve designed cared which electrons went down which branch of the circuit. I’m not sure why you do.

To answer the question that you asked to "Mr. Smartypants," think about this. Turn on the faucet in the kitchen sink. Put a knife in the falling water, so that the knife is an obstacle in the flow, but still the water (almost) all falls to the floor of the sink. What determines which side of the knife a particle water molecule will "choose" ? Obviously there will be some molecules at the middle of the stream for which the "decision" about which side of the knife to pass around, will not involve any big pressure differential. At the molecular or atomic or electronic level, there is quite a lot of random motion, and the "difficulty" of deciding which side of the knife to pass, does not prevent the water molecules from reaching the bottom of the sink.

The only difference is, as another responder noted, the average drift speed of electrons in a wire is much slower than the speed of the water molecules in my scenario.

Least resistance, or think of it as least "back pressure".

Excess electrons repel each other, so you could think of them like rubber balls being pushed through pipes; at a split, they go in whichever side is pushing back less.

"Current" is called that as it's analogous to liquid flow; like pipes. A longer/thinner pipe is comparable to a resistor.

Why do water molecules "chose" one path or another where a pipe splits? it's again whichever has least back pressure.

Each individual electron takes the path of least resistance. Every electron that goes down one branch reduces that branch's pull on the other electrons. They end up splitting up among branches in reverse ratio to the resistance of each branch.