No, it is not possible to enter the dunes from the parking lot. The dunes are a protected nature area, so you cannot enter just anywhere. You can walk back to the Van Oldenborghweg and enter the dunes from there. Here you will also find a PWN pay-station to purchase the required dunes entrance ticket. You can also purchase your dunes entrance ticket online via https://www.pwn.nl/natuur/duinkaart.