If the parent in a bridge is the root, is it a bridge cut node?
By definition, a graph that has only one vertex is connected.

That's to say if you delete A, the graph B is still connected. Clearly, A is not a cut-node because it can't disconnect the graph. So is vertex B.

A vertex is a parent/bridge cut-node if and only if it is not root vertex.

The concepts of these three types of cut-node are introduced by Skiena's in his own book, so these concepts are not very rigorous. An errata list may be a good supplement.