Definitely not! A finch has no defense against a parakeet’s hooked bill. Additionally there is the concern for possible disease from the finches, and the cage bars for the parakeet cage may not be close enough together to accommodate a finch.
Hopefully you have found a solution by now because unless the box has openings that can let in light, the finches are not likely to eat in the dark. Finches have an extremely fast metabolism and have been known to die from going without food or water for as little as four hours during the day when they are active. If it is one of those clear plastic boxes, then they should be fine for a couple of days. Otherwise you need to go out now and buy a small cage that is suitable for finches.
Thank you for asking Lafeber,