Filling gaps between polygons using QGIS

@Pil Kwon, since you have the same "post-code", therefore in QGIS I can suggest using a "Convex Hull":

Vector > Geoprocessing Tools > Convex Hull

Hint: Create Convex Hulls based on a field.

Wouldn't you want a concave hull? A convex hull will give you a large amount of area that isn't part of the block. For example, a convex hull would fill in that big gap in the "hook" park of block 1.