You are disregarding other advances in genetic engineering
When you say:
Biology tells us that sex evolved as a way to defend against viruses, bacteria, and parasites.
...you are in effect postulating that evolution through random mutations and natural selection would cease and that evolution through planned and engineered mutations, along with artificial selection would not take place. I challenge that premise. I claim that this postulate is flat out wrong. And here is why...
We are about to do not only do human cloning but also to start editing/improving human genes. So for you to assume that we will only do carbon-copy clones of ourselves instead of also improving the genome of our cloned offspring, is — I must say — kind of short-sighted.
Also, the time-span you are postulating here, where so much time has passed that we supposedly have achieved a genetically near homogeneous population, means that centuries of advances in genetic engineering will happen in the mean time. This in turn makes your assumption, that we will only do carbon copies of ourselves and never improve the genome to meet threats of the sort you are talking about, comes off as really silly.
Hence your question "How long would it take for parasites to destroy the civilization and most humans?" is made on a false premise and is entirely moot (at least for the reason stated).
When imagining scenarios like this, always remember that technology marches on.