Diodes circuit confusion
The diode with the lowest forward voltage drop will conduct and turn on while the other will turn off and not conduct current.

No, one diode will likely conduct more current than the other, but they will both conduct current.

Assuming the 2 diodes have the same voltage drop, R1 can be considered to be in parallel to R2

No, if the two diodes have the same voltage drop, the resistors will also have the same voltage drop. However, the fact that the resistors have the same voltage drop does not make them &quot;in parallel&quot;.

However in the same case, what if the 2 diodes have different forward voltage drops? Say Vd1&gt;Vd2 which assumption should I take into account?

If diodes had a fixed voltage drop (they don't - that is only an approximation) then, for each path, you would subtract that voltage drop from the total voltage across the diodes and resistors, and use Ohm's law to calculate the current in each path.
In reality, the current through a forward biased diode is nearly exactly exponentially related to the voltage across the diode. One can use a computer to solve for the current through, and the voltage across, each diode in the circuit. However, the solution technique generally involves finding successive approximations, and so is difficult for a human to calculate a precise answer by hand.

Addendum:
Why One Cannot Treat Resistors R1 and R2 in the Second Diagram as Parallel.
Suppose that

D1 and D2 are identical, and have a forward voltage drop of 0.7V
R1 has a value of 100\$\Omega\$
R2 has a value of 1k\$\Omega\$

If R1 and R2 were parallel, they could be replaced with a single resistor \$R_p\$ with a value of
$$R_p=\frac{R_1R_2}{R_1+R_2} = \frac{100,000}{1,100} = 90.91\Omega$$
and the circuit would look like



simulate this circuit &ndash; Schematic created using CircuitLab
With this arrangement, the current in each diode D1, and D2 would be identical.
However, with the actual circuit, again, assuming each diode is identical and has a voltage drop of 0.7V, we can replace the diodes with voltage sources, like this:



simulate this circuit
With this arrangement, 2.048mA flows through D1 and 20.48mA flows through D2.
Conclusion, R1 and R2 cannot be replaced by a parallel equivalent resistor, because it changes the behavior of the circuit.