Unfortunately many parents, while they may say that they love their children unconditionally, absolutely do place conditions on that love/acceptance. One must consider the parent-child dynamic; it can be complex, often rooted deeply in the unconscious. For instance, if a parent has a child who reminds him/her of himself/herself, because of shared traits, that child will carry the brunt of the parent's expectations as well as fears. And which of these depends largely upon the parent's unfulfilled dreams and ambitions, or his or her weaknesses and failures.
I think much of being a parent is actually a rediscovery of oneself, and whether one loves their children conditonally or unconditionally may depend entirely upon one's own acceptance of oneself, and whether an attitude of generosity and freedom can be extended to oneself as well.