If the parent in a bridge is the root, is it a bridge cut node? I am going through Professor Skiena's textbook and he says that there are 3 types of cut-nodes - root, bridge cut node, and parent cut node.

Now if the earliest reachable ancestor from a node is itself, it's parent and the node (if not leaf) become bridge cut nodes.

My question is that we have a simple 2-node undirected graph -> A --- B. 

In this case, B's earliest reachable ancestor is B itself and hence A becomes the bridge cut node. However, I am unsure that should be the case; since if I delete A, even then B exists and the num_components = 1?
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.