If you park in Disney theme parks parking it’s $20 for the day.
Downtown Disney parking rules:
Enjoy 3 hours of same-day parking with a $20-minimum purchase and validation from any Downtown Disney location—or up to 5 hours of same-day parking with validation from any Downtown Disney table-service restaurant or AMC Theatres movie experience.
Beyond these validations, each additional hour costs $12, charged in 30-minute increments, with a minimum $6 charge after the initial 15-minute grace period. Additionally, the maximum daily parking and lost ticket fee is $48. Parking is subject to capacity.
There is no free parking in the resort area. Some hotels offer free parking to guests but most charge a daily rate.
For DL, you can park in the parking structures. The cost is $20 per day. $35 for preferred parking. The tram that you take from the Mickey and Friends structure drops you off right in front of the World of Disney store, which is the start of the Downtown Disney district. You theoretically don't need to go to the parks to use the $20 parking. But, if you are going to Downtown Disney only, the structure isn't as convenient as the Downtown Disney lot.
Where will you be staying during your DLR visit? Some of the resort area hotels, like the Candy Cane Inn, still offer free parking to their guests. CCI is walking distance to the parks and also has a private shuttle for hotel guests. Other than staying at a hotel that offers free parking, there is no free parking nearby for DLR guests. Do not try to park on residential streets (most/all are permit zoned) or at any business with a tow-away sign. Towing is serious business around the parks and parking in a tow-away zone is not worth the risk.
While not free, another option to avoid having to park at DLR is to take Uber/Lyft or a cab. There is a drop off area on Harbor, near the tram drop off area. Depending on where you are coming from, this option can cost considerably less than the $20 per day Disney parking lot fee.
Basically, there is NO free parking at the DLR. You can get validated parking in the DTD lots with certain purchases and for a few hours only and if you 'overstay' you end up likely paying more than if you paid the $20 to park in one of the theme park lots.
All street parking surrounding the resorts is 'permit parking' only for residents. You will be towed if you park there.
Most nearby hotels charge for parking and the amount varies. Certain hotels have free parking. But either way that parking is for hotel guests only. You will be towed if you park there.
My understanding is that the nearest 'free' parking available at the ARTIC transportation hub, but it will require either a taxi or Uber to get to the parks from there. I don't know if there's an 'hours' limitation