for the following circuit could someone explain how to find the thevenin resistance across the capacitor? Most answers use the 10K resistance in parallel as the thevenin resistance but I don't understand why. Why do they neglect all the other branches, resistors and op amp?
If you're doing a static (DC) analysis to find the operating point of the circuit, then the capacitor acts as an open circuit. So the dc feedback resistance is in fact 10 kOhms, the resistance of the resistor in parallel.
If you're analyzing the AC behavior of the circuit, you should consider the impedance of the capacitor
But we don't call that the Thevenin equivalent resistance of the capacitor, we call it the impedance of the capacitor.
Why do they neglect all the other branches, resistors and op amp?
Because what you're trying to find is the resistance in the feedback path. Later you'll consider how the feedback resistor works in combination with the 20 kOhm resistor and the op-amp itself to determine the closed-loop gain of the circuit.