Several things may be going occurring in this scenario.

One, the parents stomping the cygnets are not the parents of the cygnets and they are trying to remove the rival family from the pond.

Two, the cygnets may be of the age that they are now being forced from the habitat by the parents during the migratory season so that the parents can begin the new nesting season in the upcoming months. Yes, captive swans will still feel the urge to migrate and may even walk south from their habitat. If the cygnets are moving into another family's area, the feather nipping will occur to let the cygnets know they are not welcome in the area. Have the cygnets turned white and approximately 8-10 months of age? If so, this is the most probable explanation.

Three, the parents are young swans and have no idea what the cygnets are and why something so small keeps following them around. Are the swan parents new parents this year or have they successfully bred prior to this season?

Four, something is wrong with the cygnet(s) and the parents are either trying to kill it to remove it from its misery, chase it from the family so that the other cygnets do not succumb to an illness, or the parents want to remove the sick/injured cygnet from the area so that a predator does not follow a weakened cygnet back to the family and kill the other cygnets.
If the parents are indiscriminately trying to remove all cygnets, then this probably is not the case.

So, without knowing the history of the parents and the cygnets, and which family the cygnets belong, this is a very tough situation to analyze. The Regal Swan