To pull in indicates a change of direction: “She pulled into the parking lot” -> “She turned off a particular road and into the parking lot”.
OED
7.b. Of a driver or vehicle: to drive off or to the side of the road, esp. in order to stop.
2001 T. Hanley in M. Hickey Irish Days (2004) 125 This side of Tulla there was a bit of a blind laneway and I'd pull in there and have a flask of tea.
To head = to move in a particular (usually forward) direction
OED:
20. Originally Nautical. With adverb or prepositional phrase.
a. intransitive. To move forwards in a specified direction or towards a particular place or thing. Later also more generally: to go, make one's way. Also transitive: to follow (a course, way, etc.) in a specified direction.
2008 A. Nersesian Sacrificial Circumcision of Bronx 261 Once he had located the address of the Russian embassy in the phone book, he headed over in a cab