I was just there 3 weeks ago and had no issues connecting to my flight to Napoli. The 3 terminals are connected and you can get from one to the other quite easily. Since you only have a hand luggage and an EU citizen, 60 min is more than enough. Since it's on a single ticket, they know you're coming to and connecting to a flight, so do not worry.
On www.gratisparken.de you can find a lot of free parking places in and around Düsseldorf. The closest to the airport are on these streets:
And on www.duesseldorf.de there is also a map with all official P&R in Düsseldorf.