Best residential neighborhoods

Hi! I’m visiting Barcelona with my wife and two daughters in early May. We are staying in Sitges and taking the train into the city. We love walking through cities and look forward to Barcelona. A few questions about Barcelona.

- what are the best parks to walk through

- what are the next residential neighborhoods to stroll through. Would love to find a ‘local’ restaurant in one of these neighbourhoods.

- any recommendations for ‘local’ restaurants that may not have many tourists for lunch?

Thanks

Thanks!!!!!