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They will use an interface similar to a calendar to select which times would be best for this job, and which times would work if need be. The user will click and drag over all the appropriate times to paint them with whichever color is selected - green meaning the time is good, and yellow meaning it's OK. If they accidentally color a time they didn't intend to, they can click it to erase the coloring. They will also enter a title for the job and the length of time for the job. Separately, babysitters fill out a similar interface to indicate their availability, which will be under their profile tab.
The parents then receive a list of babysitters that are available at any of their specified times, and a separate list of babysitters that indicated they could be free if necessary. They can then invite some or all of these babysitters to apply for the job.
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The babysitters receive a notification that they can apply for a job, and can apply by clicking "see details". 
They Babysitters can also view all jobs they've been invited to under their jobs tab. They can decline a job by clicking on the small X to the left of it. 
When a babysitter clicks on a job, they select the start time that is preferable to them and then applies. The parent will then receive notifications of which babysitters respond and can choose whichever they prefer.
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The parents can also leave useful information for their babysitters on their profile, so babysitters can easily look this up while on the job.
Analysis: This design is very efficient because painting over all times which work can be done very quickly. It might be a bit confusing to learn without text explaining what to do for at least the first time. It is somewhat safe since there is an easy method for undo-ing mistakes, but mistakes will be very common because any time a user accidentally drags across a time with the mouse down it will get colored.