Town drew the first-round replay 3-3 at Valley Parade on Tuesday night after scoring an injury-time equaliser to force extra time, but they then conceded in the 120th minute and the tie went to spot-kicks, which the home side won.
Illness and unavailability meant Boothroyd was able to name only five substitutes, one of them being Adebayo Akinfenwa who was so unwell he travelled to West Yorkshire separately from the rest of the squad.
The midfield quartet had shared just a handful of senior games between them and were marshalled by teenager Lewis Hornby. And the side's age got younger when Ivan Toney was introduced in the first minute of extra time at 16 years and 273 days.
Boothroyd said: "We are disappointed but I thought we showed bucket-loads of enthusiasm and bucket-loads of spirit to come back into it.
"It's probably the youngest team I think I'll ever put out and I think it was past the bedtime of one or two of them.
"I can't take anything away from the performance. I thought we were very strong and very spirited. To go out on penalties was awful."