According to Romanian Ministry of Foreign Affairs if you hold a 2 or multi-entry visa you don't need a separate visa if one holds a Schengen visa as of February 1st:
According to Emergency Government Ordinance no. 109/2013, as of February 1st 2014, the bearers of uniform visas (with 2 or multiple entries), long-term visas, as well as residence permits issued by Schengen Member States shall no longer be required to hold a short-stay visa to enter the territory of Romania for a time period that may not exceed 90 days in any 180 day period.
In order for the bearers of the aforementioned documents to benefit from the facility of entering Romania without holding a Romanian visa, the number of entries as well as the right of stay established as per the Schengen visas, must not have been exhausted.
Also, the right of stay in the territory of Romania shall not exceed the right of stay granted as per the visas/residence permits issued by Schengen Member States.
Conditions for exemption from a Romanian visa:
The validity of the above mentioned travel documents must cover the transit period.
The above-mentioned citizens are exempt from the Romanian transit visa, but are not exempt from the obligation of fulfilling the general conditions of entry, in accordance with the provisions of Article 5 (1) from Regulation 562/2006 of the European Parliament and of the European Council, of 15 March 2006 – The Schengen Borders Code (they must hold a valid travel document, supporting documents attesting the purpose and conditions of the trip, as well as be able to make the proof of appropriate financial means for the duration of transit);
In order for a uniform visa, a national visa, or a residence permit to be equivalent to the Romanian transit visa, under the conditions listed above, it is imperative that the transit through the territory of Romania naturally fit into a coherent itinerary to the country that issued that visa or residence permit, or into a return trip to their home country or country of residence.
Holders of a single-entry visa, who have exhausted their right of stay, are allowed to transit Romania exclusively for the period of validity of that visa.
**Holders of permanent residence permits, issued by member states of the European Union, may enter and reside the territory of Romania for a period of up to 90 days, within 6 months, without a Romanian visa.
So long story short. Since Wizz Air in your case doesn't facilitate transfer and you need to re check-in for the flight from Bucharest to Paris you will need to get a visa since Transit Without Visa(TWOV) is intended for passengers staying in international transit area, which you do not.