> Why would a parasite waste all of the time to become host specific if the goal was to kill off the host? 

Evolution does not project into the future, it is the result of millions of years of adaptation. It has no goal "in mind," and especially organisms like parasites have no goals, no concepts of hosts, nor of specificity. Parasites become host specific over time by isolation, genetic drift and natural selection -- not by some sort of goal oriented self evolution. Even the very idea of species is a human created concept, life is an interconnected system. 

In any case, a parasite killing an individual organism is not "killing the host"  -- they do this all the time, and have plenty of mechanisms in place to move on to another host. Finally, if a parasite were to kill all susceptible individuals, it would simply go extinct. Which is what happened to the majority of the world's life forms. There simply is no guarantee that any species will survive forever. 

Finally, varroa were introduced to Santa Cruz Island off California for the sole purpose of eradicating honey bees from a habitat where they were considered "non-native." Parasites DO kill their hosts, and themselves at the same time:

> introduction of a parasitic mite as a biological control (1994–1998), located the last known feral colony at the top of the Matanza grade on August 6, 2002 and found that same colony dead on March 31, 2003. 

See
WENNER, A. M., THORP, R. W., & BARTHELL, J. F. (2009). Biological control and eradication of feral honey bee colonies on Santa Cruz Island, California: a summary. In Proceedings of the Seventh California Islands Symposium. Institute for Wildlife Studies, Arcata, California, USA.

PLB

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