Living in a conservation area does not, on its own, stop you fitting roof panels. The law singles out walls and enclosures that front a road, and councils can go further with an Article 4 direction.
In England, roof-mounted solar on a house in a conservation area is normally permitted development, within the same 0.2 metre (pitched roof) and 0.6 metre (flat roof) limits as anywhere else. What is excluded is equipment on a wall, balcony or roof enclosure that fronts a highway. An Article 4 direction or a listed building changes this, so check with your council first.