Assuming the capacitor was not over sized to begin with, ripple will increase with load.
The capacitor is there as a storage device. It charges as during the leading edge of the sine wave and will discharge into the load on the falling edge. Ripple will depend on the size of the load. If the capacitor is of sufficient size, you may not notice a significant difference in ripple when you put the second 1000 ohm resistor in parallel with the first. If the capacitor was not oversize to begin with, then yes. you will see ripple increase.
The following shows he falling edge where the capacitor supplies current. The heavier the load, the more the capacitor will discharge on the falling edge, the greater the ripple and the lower the average voltage.
So yes, the load will effect ripple and average voltage.