Why could my electrolytic capacitor have current but not my parallel plate capacitor?
With most off the shelf meters, the meter won't detect much current because the capacitor essentially gets shorted when the meter is placed across it.
With a capacitor the exponential timeconstant (~60% of the initial voltage) would be
$$ \tau = RC$$
With a meter in current mode most likely having a resistance of lower than 0.1Ω
$$ \tau = (0.1Ω)(1000uF)= 100us$$
So this means in about 100us most of the voltage fades away

Source: http://hyperphysics.phy-astr.gsu.edu/hbase/electric/imgele/capdis.gif
The current could also be calculated with the equation above, but you must know how much voltage \$V_0\$ you charged the capacitor to.
The current at 100us and 5V would be less than 19mA, in another few hundred us (like 1000us) the current from a 1000uF cap would be less than 1mA which would be hard to see for many meters because of the short duration of the current.
The capacitor you constructed probably has a smaller value of capacitance (maybe lower than 1uf) so the time would be even shorter than 100us.
So, one way to overcome this would be to put a resistor in series with the capacitor to make the time constant longer and/or use a really good current meter that can measure. Another good capability to have would be graphing. Don't get me started on fitting exponential curves (big hint).