Finding time constant in terms of R and C
The following steps are the transformations that are probably easiest to see, in order:







simulate this circuit &ndash; Schematic created using CircuitLab

You did the first step, correctly. That was just two resistors obviously in parallel with each other. That's Step 1 shown above.

However, for the following step you don't add \$R_1\$. The reason is that once you've reached the upper-right corner schematic, you are left with a voltage divider pair of resistors. And this means you need to perform a "source transformation."

To help make the transformation a little clearer to discuss, I decided to ground the (+) rail. The reason is that \$R_1\$ is in parallel with \$C_1\$ and grounding the top rail allows me to flip the schematic over (upside down, from the voltage perspective) to make the resistor divider analysis a little more obvious. That's Step 2, above.

A Thevenin source transformation of a voltage source and a resistor divider pair is pretty simple. In your case, the new equivalent voltage source is \$V_\text{TH}=V_\text{source}\cdot\frac{R_1}{R_1+\left(R_2\mid\mid R_3\right)}\$, where \$V_\text{source}=-V_1\$, and its source impedance is \$R_\text{TH}=\frac{R_1\cdot\left(R_2\mid\mid R_3\right)}{R_1+\left(R_2\mid\mid R_3\right)}\$; this latter calculation simply being \$R_1\$ in parallel with \$R_2\mid\mid R_3\$.

If you haven't been exposed to the Thevenin source transformation for a voltage source and a resistor divider pair, as yet, then you should immediately work on that problem and make sure you understand why it works as it does.