Why does Sosuke call his parents by their first names?

As far as I can tell, Miyazaki has not commented on this, so all that's left is speculation, and people have been speculating since 2008 online on this subject. As you note, in Japan, addressing someone by their first name indicates great familiarity or contempt, and this is an unusual thing for a child to do with their parent.

1. Desprezo is a possibility. With his father often out on military ventures, and his mother leaving him alone to take care of the older people, one could imagine resentment manifesting in disrespectful address. That said, despite Miyazaki's occasionally strained relationship with his son, his films generally feature strong familial bonds, so I think this is unlikely.
2. Familiaridade is also quite possible. Sosuke and his mother evidently work closely together, and it may be that this very informal manner of address is an indication of their close and loving relationship.
3. Respeito is a third possibility. With Sosuke's father often out on maneuvers, he seems to often play the role of the "man of the house". Therefore, his mother allowing (or encouraging) the informal address is a matter of respecting a degree of equality.
4. Modern/Western family patterns is a fourth possibility I've seen suggested, that Sosuke's parents are being influenced by Western culture, and adopting some of those practices by allowing this degree of informality, treating their son as a colleague and friend instead of a subordinate.

Ultimately, I think that the most likely answer is respect, and a recognition that they allow Sosuke to be independent to a degree not all children enjoy.