Many analysts said the approval had been seen as a formality but signs the Greek bailout was on track boosted the euro zone common currency.
Further gains could be capped if a stronger dollar trend driven by higher Treasury yields reasserts itself. The euro could run out of steam above $1.33, especially if euro zone purchasing managers surveys on Thursday come in weak.
The euro rose to an almost two-week high of $1.3284, according to Reuters data, after Greece's lawmakers approved the country's second bailout deal, as expected. It was last little changed at $1.3224.
Greek parliamentary support for the new 130-billion-euro bailout package is encouraging, said Camilla Sutton, chief currency strategist at Scotia Capital in Toronto. Tomorrow's PMIs will be important, as recent data has pointed to some stability in several European economies and the PMI will help confirm this, which would be positive for the euro.
Technical analysts said the euro rally was likely to run out of steam around $1.33, just above the 61.8 percent retracement of its late February to mid-March fall.
There has been an easing in general concerns about euro zone liquidity and the creditworthiness of euro zone banks, plus euro short positions can carry on being unwound, said Adrian Schmidt, currency strategist at Lloyds in London.
He saw potential for the euro to rise towards $1.35 against the dollar, around the top of its recent range, although short-term resistance at $1.3320 may prove too high a hurdle if Thursday's preliminary PMI data comes in weak.
Investors remained wary of the risk of another flare-up in the euro zone debt crisis. Greece got its first batch of bailout payments this week, but the Italian government looked set to clash with unions over employment law reforms.
A firmer dollar, due to fading expectations of more monetary easing by the U.S. Federal Reserve after a modest brightening of its economic outlook, may also hamper the euro.
Ten-year U.S. Treasury yields were last trading around 2.3393 percent, within sight of a near five-month high touched this week.
U.S. housing data, due at 10 a.m. EDT (1400 GMT), could boost the dollar if it adds to expectations that growth in the world's largest economy is picking up.
WEAK YEN, UK BUDGET
Rising U.S. yields and monetary easing from the Bank of Japan last month have boosted the dollar, particularly against the yen. The greenback rose 0.4 percent to 84.02 yen, just shy of an 11-month high touched last week.
During the Asian session Japanese exporters were seen selling the dollar ahead of the end of their financial year on March 31, but market players said there was good demand to buy the greenback on dips.
The euro also hit a near five-month peak of 111.43 yen, according to Reuters data, nearing resistance around the peak hit on October 31, when Japanese authorities last intervened in the market. It was last at 111.07, up 0.3 percent.