This is unlikely to be a significant negative for the dollar. They're unlikely to start moving heavily out of dollars as U.S. assets still yield more than in Europe or other countries.
The potential for the euro to fall is pretty limited from here. The dollar is not going to get the same support we saw last year from rates now, and the euro zone economy is looking more solid.
On a short-term trading view the euro looks pretty awful,
All year the euro has been damaged by comments emanating from German politicians,
The euro has lost ground against every other currency, apart from the Brazilian real and a few eastern European ones,
It highlights that a great deal of investment is flowing out of the euro area and into the U.S., the U.K. and elsewhere,
If everything remains the same, the euro should be closer to $1.08 than $1.02,