It depends on how you make the measurement.
Lets stop and look at your calculation there a little. You've shown that a perfect inductor in series with a perfect resistor has the same impedance as a perfect inductor in parallel with a (different) perfect resistor.
Any real measurement system measures impedance. A single impedance measurement doesn't tell you whether you're measuring a parallel or series arrangement. If you're making a measurement with an LCR meter, you can normally choose to have the measurement displayed in one of several ways: a complex impedance; or as a resistance and capacitance/inductance. If you select the second option, then the meter works out the resistance using a series or parallel model, and you, the operator, have to tell it which model to use.
What about an handheld mulitmeter? Well, they have only one frequency they can measure at, 0Hz, also known as DC. If you look back at your maths, then when \$\omega\$ is zero, \$Rs=Rp\$, and that is the number which will appear on the meter.
So how can you tell the difference between a series and parallel system? You have to take more than one measurement, at different frequencies. If you take several impedance measurements, then calculate the inductance and resistance using one model, and get the same result each time then you have chosen the right model. If you get different results, then try the other model. If neither model fits, then the system you are measuring is more complicated than just two components.
For a real indictor, the model of a perfect inductor in series with a perfect resistor is pretty good. For a real capacitor, the model of a perfect capacitor in paralell with a perfect resistor is good. If you measure up to very high frequencies, then you'll need a more complicated model to accurately represent the real component.