Parker said the NLGN was succeeding in 'stirring up the issues so that all political parties now bite into the localism agenda'.
The group is funded by 26 local authorities as well as a number of corporate partners. It has been a strong proponent of directly elected mayors and city academies.
Parker's successor will oversee the group's press strategy, parliamentary affairs, conference programme and management of its publications. The NLGN has also established an 'innovation network', which it bills as 'a space for ambitious local authorities to experiment with new ideas, share learning and influence public policy across all levels of government.'
Parker will join LLM on 13 March as an account director, reporting to the consultancy's managing director Craig Leviton.
Leviton said Parker would work across 'local government, regeneration and housing'.
The firm represents several councils across England including Blackpool, Hounslow, Liverpool, Manchester and Nottingham.
Prior to the NLGN, Parker worked for two years at left-leaning think-tank the Social Market Foundation and for Peter Kilfoyle MP.