How to prevent voltage variation in a fast-switching circuit with parallel connection?
You have not given enough information in your question to provide a good answer. 

The 10 Hz signal you are seeing is the "beat" or difference frequency between the two frequencies you are generating. This is a problem because your supply is not "stiff" enough. The voltage is drooping as current increases.

From Ohm's law we can work out that the resistance between the first and second LED is \$ R = \frac {V}{I} = \frac {0.2}{0.2} = 1 \; \Omega \$. This seems reasonable for a breadboard.

Try wiring the positive and negative of each of your circuits directly back to the power supply. (Don't use the common rail of your breadboard.) This "star" arrangement will avoid each circuit interfering with its neighbours.







simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1 R1 to 12 represent the wiring and contact resistance. (a) In this circuit the loads on the right will cause interference on the loads to the left. (b) In this circuit each load is powered directly from the power supply.

In Figure 1b there will be no fluctuation in lamp voltage if the power supply is well regulated.