In Kohut's self-psychology, the notion of the parents as an infant's selfobjects comes close to supposing this. So one answer might be 'self-psychology'.
Kohut avoids defining the self, so it is impossible to know whether he supposes the self would exist without them, but the parents are not conceived of by Kohut as truly separate from the self of the infant early in development. They are its 'selfobjects', the things out of which its 'self' is really made.
Going beyond Klein's notion that an infant introjects the world, pulling important objects into itself in phantasy to be rid of the anxiety about their absence, Kohut sees the self itself as truly made up of attachments to objects (mostly people, despite the word).
In the same way a transitional object like a favorite stuffed toy can become a replacement for continual holding, another object can take the place of a selfobject out of necessity, but the need for some connection back to the original remains real. (At some point Mommy can set him down at will, but if the 'wubby' is lost, the child falls apart.)
In the most direct application of the theory, for an addict, the substance and the feeling of being in an altered state of mind can be selfobjects that have replaced earlier selfobjects like maternal regard and the giddiness of being regarded.
Deprived of selfobjects, one's self lacks cohesion and one 'fragments'. Narcissistically healthy people have a broad enough set of associations to selfobjects and clear enough internal images to evoke those associations, that they are never truly deprived of them. Though even healthy people, placed in a foreign enough environment, may begin to fragment due to a lack of reminders and experiences of their selfobjects.
(Of course this has to do with selfhood, not existence. Someone could exist entirely as an extension of another person, with a common self. It seems to me that we do so in the womb and continue to do so for a while as infants. Of course that does not mean that fetuses or infants have no physical existence.)