Is it safe to mop a parquet wooden floor, sealed in polyurethane, with a damp mop ? Will it damage the wood ?

Damp MEANS not dripping wet.  And you will be fine.

sponge mop w murphys oil soap,a cap in a bucket,w a dash of tide laundry soap in the yellow bottle,,run the mop under the sink hot water tap squeeze and rinse a few times

A damp mop won't hurt it.

Vinegar and water work really well. My floors were just refinished and coated with poly--and that's what the guys told me to use. Just don't let the water SIT on the floor. 

Polyurethane is meant to protect the wood--so no, it won't damage them. 

You can also use a Swiffer mop. 

I use wood floor cleaner and a damp mop. Do not use too much of the cleaner coz it can be slippery when dry. 

We, too have parquet with a polyurethane finish that was literally poured on.  That surface should be water-tight - but you absolutely don't want to let water penetrate around edges or through joints:  the pieces of wood will swell and pop-up and you can never repair the floor invisibly.  

For years, we used a slightly damp mop on ours - it began to look dull and dirty.  We found a Pledge product:  "Pledge Floor Care 4-in-1 Wood Cleaner" . . . . WoW:  dust the floor, then apply a light coat of this stuff with a damp mop.  No buffing - it dries crystal clear hides scuffs and scrapes and SHINES like crazy.  We do it no more than once a year - usually 2 years - and just dust mop in-between.  It has cut maintenance on the floors to just about nothing.

I understand that they changed the name of the product to "all in one" or some-such . . . same stuff.  NO - this isn't spam - I don't work for Pledge - it's just a great product !!!!!

yes it is as long as it is not permanently wet it will be OK

Not quite sure what a sealer is, are you?

Its fine, just damp, not soaking wet.

How would it damage the wood if it's actually sealed?