RFC5639 defines some curve parameter for Elliptic curve cryptography.
Aren't there any curve parameter database for Hyperelliptic curve cryptography? What I can only find was that written in this PDF.
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There is no database of hyperelliptic curve parameters because their use is not sufficiently popular. This is because the topic is too new and the advantage of faster calculations doesn't outweigh the disadvantages of novelty and lack of reference implementations.
Another aspect is that with modern computers and algorithms, anyone seriously looking at implementing hyperelliptic curve cryptography can easily generate their own parameters. Contrast this with the early days of elliptic curve cryptography where finding (let's say) a twist-secure prime-order curve of a decent size was a significant computational task.
As soon as hyperelliptic cryptography becomes popular then there will be databases of parameters to ensure interoperability between different implementations. However, I don't anticipate this will happen for a long time because groups of points on elliptic curves are a sufficiently good improvement over groups of integers for cryptographic purposes.
The slides you link to and their paper on the eprint archive are excellent and you should be able to use their parameters with some confidence.