OK, here is some info on mapping and the differences.
1) RAM use
For a tile-based map, you need one number (usually 2 Bytes for tilesets with more than 256 different tiles) per layer to identify the entire tile, and then the data on the tileset itself.
For a picture-based map like a parallax map, you need 3 Bytes per pixel (24 Bit Color) or more for truecolor pictures. A tile consists of 48x48 pixels (MV - or 32x32 for Ace and older), that means for the same area of a single tile the parallax map in MV needs 48x48x3 = 6912 Bytes, again per layer. It does not need tileset data, but that is sometimes still loaded for other purposes.
A tilebased map needs more layers however, but that still means that a parallax usually requires several thousand times the RAM compared to a tilebased map.
And here is another catch: The filesize you see when saving your parallax - that is compressed data.
To work with a map in RAM, it cannot be compressed.
Make a test - store your parallax as BMP instead of PNG. That is a file format that has no compression. So the filesize of a BMP tells you how much RAM the same map needs while used in the game...
2) Map design
This one is on the developer and has nothing to do with the difference between parallax and tilebased mapping:
In order to create good maps, you need to be able to design them correctly - to know what to place where and when.
There are a lot of very nice looking maps out there that are absolutely unplayable because the mapper couldn't design them correctly.
Simply because the nice look resulted in an unplayable labyrinth of obstacles for a map that the player has to pass hundreds of times while playing.
3) workload
A parallax map always requires much more work to create than a tilebased map.
You can get faster with experience, but you can't change that a parallax map needs several additional steps that are automatic with a tile map - things like setting passabilities. And more placement options (non-grid) means more work when placing.
When I started here and asked around, one of the artists here gave me this guideline:
A tilebased map is 10% the work for 90% the quality of a parallax map
Yes, a parallax map looks better - but if you know your map designs, then you can usually create five to ten regular maps that are almost as good-looking in the same time you create a single parallaxed map. And that is a lot of time saved for single person game developer with a game that has several dozen maps.
However, that all depends on the developer knowing what to do on maps - and that is why I placed the point about map design before this.