City of York Council states that "there are 35 conservation areas in York" and that "Conservation areas have extra planning controls applied to them to help preserve or enhance their character and protect their settings". Inside one, planning permission is needed to "clad walls", "in most cases, unless repairing or renewing an existing surface", and planning permission and / or listed building consent is required to "substantially alter or build a new structure/boundary", with a heritage statement also required in that instance. Which consent applies is a matter to check with the council.
The council's Article 4 page states that its directions remove "permitted development rights for certain minor alterations, extensions and changes of use" where planning permission would not normally be required, and it names East Mount Road, Heslington, houses in multiple occupation, The Punch Bowl public house and 79 Fulford Road. It does not mention painting.
The national wording behind all of that is the Planning Portal's: "You do not need to apply for planning permission for repairs, maintenance or minor improvements, such as painting your house", while in a Conservation Area, a National Park, an Area of Outstanding Natural Beauty or the Broads "you will need to apply for planning permission before cladding the outside of your house with stone, artificial stone, pebble dash, render, timber, plastic or tiles". The Portal adds that "If you live in a listed building, you will need listed building consent for any significant works whether internal or external".
We would explain how you check the conservation boundary, any Article 4 direction, the listed building position and the proposed work with City of York Council before confirming whether permission is needed.