"I'm a skeptic on that for the time being. I think the Fed is going to come out in December and do a one and done, and if it does, that's going to be a cap on the dollar rally," said Boris Schlossberg of BK Asset Management. "The euro is under a lot of pressure, but it's going to test the 1.05 level before it gets to parity."
Schlossberg said the dollar's rise could also be slowed by weakish economic data, which would indicate a slower trajectory for Fed rate hikes.
"What we learned this year is the Fed is sensitive to the dollar, so I think the dollar is very strong," said Vassili Serebriakov, currency strategist at BNP Paribas. "They'll hike in December but the tone would be more dovish. ... The ECB is probably less sensitive to the euro. If the euro is under $1.05, they'll ease but the easing, the mix of the measures they choose, would be less aggressive. All is well for the U.S. dollar, but I think we just have to be careful as to the speed with which it does move higher."
Read MoreTalk of euro parity with US dollar heats up
The dollar could gain further if the Fed raises rates Dec. 16. That move will come on the heels of a European Central Bank meeting Dec. 3, where it is widely expected to take further easing actions, including a possible rate cut — a negative for the euro.
While the dollar is expected to move against a whole group of currencies, particularly those in the emerging world, its move against the euro comes against the backdrop of two major central banks moving in polar opposite directions on policy. If the Fed acts in December, it will be ending a seven-year policy of zero rates.
"It's an unusual event to have these central banks moving in opposite directions," said Alan Ruskin, head of G-10 currency strategy at Deutsche Bank.
Serebriakov said he expects the euro to reach parity but not until the middle of next year, and Ruskin and others also believe it will take awhile.
"It's hard to get those big moves on the dollar. The dollar is getting quite expensive at this level," he said.
Read More'Perfect storm' to lead to euro-dollar parity by December?
Marc Chandler, chief currency strategist at Brown Brothers Harriman, expects the dollar's gains against the euro to slow, particularly after the common currency's swift decline from $1.14 in mid-October. The euro was trading around $1.07 Wednesday, and the dollar index was off slightly.
Chandler said the dollar's rise is nowhere near an end, and the divergence in central bank policy is far from peaking. He said the the euro could even fall back to test its 2000 low close to 82 U.S. cents.
Read MoreFed rate hike won't stop bull market run: Expert
"If the Fed raises rates in December, they may not raise them again until Q2. I am a big dollar bull but I think the price action itself might be frustrating if you're looking for another swoosh down (in the euro)," said Chandler.
Ruskin said that interest rate spreads are moving in favor of the dollar and markets are pricing expectations for a firming dollar for several more years.