Electrical – Resistor Network with voltage source and current source

There is no return path from ground back to the other side of the source, therefore no current will flow to ground. All of the current will flow through the resistor.

In your equivalent resistance equations, you are missing the fact that a voltage source has impedance of 0, and a current source has infinite impedance. Anything in parallel with 0 impedance still results in 0 impedance. Likewise, anything in series with infinite impedance still results in infinite impedance.

To help convince yourself, try putting some real number on your top example. Pick some values for the voltage, Rp, and R. Now put different loads on the result and compute what the voltage on the load is with and without Rp. You will see that Rp doesn't matter. The voltage across the voltage source is the specified voltage, by definition of what a voltage source is. Put another way, a ideal 10 V source always gives you 10 V, whether someone else is drawing 10 mA, 10 A, or nothing from the same voltage source.