Perhaps there are some people who will think that holding an umbrella is not a difficult enough task to require a special hands-free model, but it makes sense if you have to carry a baby (or two, Phinnaeus? Hazel? Seriously, Julia?) in your arms in the pouring rain. It's also a lot less unsightly than one of those umbrella hats.

Or the umbrella can be attached directly to your little one.

The hands-free umbrella appears only to be available in Japan right now. Thanks Dennis!
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The umbrella, called the Kataburera (shoulder umbrella) was developed by holistic physician Daryn Peterson not only for people carrying lots of bags and/or children, but for the elderly and infirm, who are prone to accidents in bad weather. The Kataburera is extremely lightweight, so it shouldn't affect balance or put too much pressure on fragile bodies. You can read more about the umbrella and Daryn Peterson in the Japan Times.
The company's website is kanarazunaoru.com (in Japanese). Peterson is seeking to distribute world-wide but so far the umbrella appears only to be available in Japan.