@Sandor: Thanks for the answer. I am not familiar with all the terminology here, but this only seems to show that the particular ideal does not satisfy the desired condition. I don't see how it shows there are no other prime ideals satisfying the condition in a regular local ring apart from the family mentioned in the question.

@Karl: Thanks for your response. Does this really give us a new example? Since, $R$ is regular, $R/I$ is regular if and only if $I$ is generated by a regular system of parameters. This takes us back to the family I mention. Am I missing something here?