"Δεν αφήνουμε τα παιδιά μας να πάνε μόνα τους."
Μετάφραση:We do not let our children go on their own.
How about "The kids want to go to the park but we don't let them go on their own". Does this meaning work?
Yep, that would work! Another that would work would be "We don't let the kids go to the park on their own." In your example, the second "to the park" is implied by the first.
The same holds for "by themselves": "We don't let our children go by themselves" might work in some context but not by itself.