I quote from the excelent analysis made by RBS from "RBSM: Global Currency Themes for the Week | Participation Rates" :
"The Quantity of Money | When considering short-term EUR/USD valuations it's important to look beyond relative yields. Yields have moved steadily against EUR/USD over the past year, yet spot has continued to move higher. While spread compression in the periphery is an important driver, we believe that the quantity of money is also important. With rates at the zero lower bound, relative monetary conditions can't be characterised by rate spreads alone. Relative central bank balance sheets also need to be considered. While the Fed is still easing through QE, the ECB is reducing the amount of EURs in the system. Excess liquidity levels have dropped below EUR 100bn. It means that the ECB balance sheet (versus say the Fed or BOJ) is shrinking which tends to be EUR/USD supportive. "
Add to the above the fact that the Euro zone have a positive current account and voila, EURO is very well bid against USD.