Some of those Lipo voltmeters are not very accurate. I would (carefully) confirm the cell voltages with a voltmeter. Measure cell to cell.

I parallel charge all the time, never had any issues with cells getting out of balance.

If your parallel board has an LED on it that lights up when you plug in a battery, remove it.
They connect it to one cell, and it will ( if you leave the packs connected) bring down that set of cells.

Haven't  received it yet but my solution to the problem is a Thunder Power 4 unit charger.  It has a AC/DC  input. 

http://www.thunderpowerrc.com/CHARGERS_POWER_SUPPLIES/Chargers_2/TP4X610HV

MORE DETAILS
AC Input voltage   110Vor220V
DC Input Voltage   11-18V
Charge power   max.100W*4
Discharger power   max.10W*4
Charge current range   0.1-10.0A
Discharge current range   0.1-5.0A
Balance tolerence   ±0.01V
NiCd/NiMH battery cell count   1-15
LiPo/Li-ion/LiFe battery cell count   1-6
Pb battery voltage   2V-24V
Digital Power   3V-24V

If I read the above set of specks correctly I can charge four 4-cell lipos at 6 amps each.  ie 14.4 volts times 6 amps = 87W.  This is below the 100W limit for each channel.   Also with this setup I can can charge any mix of batteries because each of the four channels are independent.  On sale at $135 it appears to be a cheep solution for parallel charging.  I have eight 4 cell LiPo batteries I plan to use this summer to learn how to fly the beginner's patter. When I get home I can storage charge or full charge all eight in two charging sessions.  If I am in a hurry at 4 to 6 amps it won't take long. 

The online manual stated that the input current is 1 - 25 amps software limited.  Output charge current is limited to 20 amps and/or 250 watts at 12V input, 500 watts at 24V input. 

Your 10 amps output took 20.5 amps input at about 12V assumed, or 246 watts, still all in limits.

If the Lipo meter is one of the inexpensive field units, they are not always accurate, as noted.  Put a pack on charge by itself and turn on the charger and check balance immediately as it comes on, not giving it time to "re-balance."  Your nice charger is likely more accurate that the cell checker.  Or with a digital voltmeter cell by cell as noted above.

As noted the charger sees 5 cells.  Each single "cell" is a composite of the cells in parallel.  Charger Cell 1 is pack cell 1 of each pack.  The charger should balance Charger cells 1 - 5 and not the separate pack cells. They are however connected in parallel and should be at the same voltage.

You can make a setting on the charger that stops at an arbitrary voltage that you select.  At about 4.13V/cell, I will put an additional 200 - 250 mah in each battery to finish to 4.2V/cell.  Parallel charge to that voltage, your choice of charge rate.  Either not balance, or balance and set the termination C rate to 5 or less.  To finish charging, balance charge each pack at C/10 or C/20.  You get reasonable fast parallel charging and then slow but reliable balance of each pack.  That said, when my packs are good and reasonable similar in IR they parallel charge fine, including Zippys.  Packs not staying in balance should be watched closely.

Your note about new TP batteries -  "They were purchased at the same time and had the storage voltage at the moment I started charging. The storage voltage was the same within 0.1 Volts: three were showing 18.9 V and one 19.0 V when I used the LiPo voltmeter just before charging" - makes me wonder if was done at first cycle.  My batteries changed IR and balance after a soft charge and discharge, and then balanced and had fairly equal and low IR.  TP suggests break in cycle, perhaps it helps life span by balancing.

Was the imbalance after the cycles, or at the onset?

Most inexpensive lipo cell checkers are also cheaper and less accurate electronics from what I read, not just decimal places.

If you want to try the charging mentioned, use one of the blank presets to make up your specifics - end charge voltage, charge rate, end C termination....  That way you can partially charge the batteries using a preset, then finish on a preset that goes to full charge.  C/5 will give slightly less capacity then C/20, but is faster.  If you don't need every last electron, use C/X and X less that 20.

Hi Motorman,
I am waiting for the Revolectrix customer support reaction.
They stay mute so I gather I confused them with my questions.

My 5S 2800 TP batteries have been cycled four times at the beginning of their use using the balanced charge from 3.80 to 4.20 Volts and then discharge to 3.80 V/cell using the 150 Watts, Made-in-China, 3 halogen 50 W light bulbs, discharger. The discharging current during these four cycles was about 5 Amps.

Later,
M