Under Trump there is one plausible scenario of EUR trading under parity - BTMU

Analysts at Bank of Tokyo Mitsubishi explained that analysing EUR/USD moves since trading began in 1999 shows that the current EUR/USD drop (since mid-2014) is the largest not to be followed by a notable rebound since the period between 1999-2002 when EUR trading began and EUR/USD dropped by 30% to the all-time intra-day low of 0.8230 in October 2000.

Key Quotes:

"The consolidation period back then is also very similar in length of time to the current period of consolidation (around 19mths). Back then, G7 intervention in 2000 to
support EUR along with improved confidence amongst central banks in holding EUR in reserves fuelled the start of a 3-year uninterrupted period of EUR appreciation
(2002-2004).

In the current scenario, we see greater risk of the next phase in EUR/USD direction as being a further lurch lower. If the US reflation scenario becomes reality, coupled with the political risk premium, we can easily envisage a scenario of EUR/USD trading below parity in 2017."