Not only Dutch tourists, but also Belgian non-Antwerpians will no longer be allowed to park in the city centre. With the new rules, the city council wants to make the historic city centre more appealing AND give the city's residents more room.
On the outskirts of the city, however, there are car parks where visitors can park their cars. The municipality of Antwerp's website gives options on how and where visitors can park and then travel to the city centre by public transport.
From the Netherlands, travelling to Antwerp city centre is very easy by rail. The train is the smartest option.
You can book a single ticket to Antwerp starting from € 21 with Intercity Brussels. If you book at least 3 days in advance, you can benefit from the advantageous Early Bird rate. Availability is limited. If you have a subscription with 100% discount in the Netherlands, you will be pay even less.
If you want to travel faster, you can opt for Thalys. The earlier you book, the lower the price. The cheapest tickets cost € 29 for a one-way ticket.