As an organisation (aero club) approved under the national legislation of Member State X, the national legislation of State X applies to you, to the skydivers, to the tandem passengers and all other registered members. Reg. (EU) No 1178/2011 on aircrew is not applicable to aero clubs that do not provide training for one of the Part-FCL licences and ratings — LAPL, PPL, CPL or ATPL.
The operation of the aircraft must be performed in accordance with Part-NCO of Reg. (EU) No 965/2012 on air operations, as the aircraft you operate is a non-complex aircraft.
The competent authority for the oversight of your dry-leased aircraft is the State of registry, that is, the state where your aircraft is registered (see Reg. (EU) No 965/2012, NCO.GEN.100 ‘Competent authority’). However, the competent authority of Member State Y may delegate its oversight tasks to the competent authority of Member State X.