What is the opposite of a dependency?  In terms of what word best
describes the relationship in reverse ie: parent to child.

Hello and thank you for your question.  The opposite of dependency in
the parent-child context is paternalism.  This is true literally
("pater") and also in common usage.

pa·ter·nal·ism
A policy or practice of treating or governing people in a fatherly
manner, especially by providing for their needs without giving them
rights or responsibilities.
http://www.thefreedictionary.com/paternalism

paternalism 
The attitude (of a person or a government) that subordinates should be
controlled in a fatherly way for their own good
http://wordnet.princeton.edu/perl/webwn?s=paternalism

Paternalism refers usually to an attitude or a policy stemming from
the hierarchic pattern of a family based on patriarchy, that is, there
is a figurehead (the father, pater in Latin) that makes decisions on
behalf of others (the "children") for their own good, even if this is
contrary to their opinions.
It is implied that the fatherly figure is wiser than and acts in the
best interest of its protected figures.
http://en.wikipedia.org/wiki/Paternalism

ANTEBELLUM SLAVERY
PATERNALISM
To understand the models of paternalism you must understand the
slaveholder/slave model. Under this model, the slaves were dependent
upon slaveholders because slaveholders could not be responsible for
defining and promoting their own goods. This argument does not rest on
the motives of the slaveholders. The argument is that it could be true
even if all slaveholders were motivated to hold slaves because they
felt it would promote their own good rather than the good of the
slaves. Many of the early supporters of this model assumed the natural
inferiority of slaves. Slaves were thought to be subhuman creatures
that could not survive in civilized societies.
....
The parent/child model is the most common model, which represents a
clear example of paternalistic interference. Many paternalistic
accounts of slavery draw upon this model. In this model, the
slaveholder is viewed as the parent and the slave as the child or
childlike.
http://cghs.dadeschools.net/slavery/antebellum_slavery/paternalism/five_models.htm

To cite an example, one of the important principles governing Japanese
behavior patterns is the ethic of reciprocity, fostered by
interpersonal dependency and a sensitivity towards the state of
belonging. Whereas in the West, dependency is viewed as being
tantamount to subordination or even inferiority and therefore tends to
be suppressed and relegated to the subconscious. In the Far East
island nation of Japan, sensibility to the need of dependency has not
only survived without having been negated but can even be said to be
institutionalized within the social structure. The "belonging" that is
the psychological foundation of the parent-child relationship and the
primary model for human relationships is referred to as amae in
Japanese.
http://www.fww.org/articles/congres1/amorita.htm


Search terms used:
paternalism definition
paternalism parent child dependency

Thanks again for bringing us your question.

Google Answers Researcher
Richard-ga

In general, I would say that the best antonym of "dependency" is "autonomy."

In your example, if the relation of the child to a parent is a
"dependency," you could call the relation of a parent to a child a
"custodianship" or "guardianship."

I would say that the most general opposite to "dependency" in your
sense of the word "opposite" would be "authority". However, I agree
with Pinkfreud that the true opposite is "autonomy" or "independency"
(the latter is a word, though little used).

What about "supremacy"?

I like "support", in the noun form.  A dependent depends on a support.
 A support has things dependent on it.  Works for me.