As I wrote, the rotating "European Presidency" is held by a country, not by an individual president. Also that is only the presidency of the Council of the European Union, which is the meeting of the departmental ministers (e.g. ministers of economy, the ministers of environment, or ministers of science policy of the member states - of note is that the heads of the governments of the member states do not belong to this since they are in the European Council unless they also were the hat of a specific departmental ministry), separate from the president of the European Council, the president of the European Commission and the president of the European Parliament - all of which are sometimes incorrectly referred to as "European president".
We can discuss on the exact amount of power of the European Parliament, but they have no right to take the initiative on European legislation: the Parliament can only vote on proposals of the Commission. The European Parliament has the right to request the Commission to initiate a legislative procedure and put forward a proposal, and they only have power of the Commission by adopting a motion of censure - which has never happened. The Parliament has no power over the European Council and over the Council of the European Union which both represent the member states.