It is a very personal decision to bring a homecare caregiver into your home.  You and your parent should determine that the company and the person that will be supplying services is the right one for you.  Take your time in an initial telephone interview.  Don't hesitate to ask any questions that come up, including hypothetical ones.  Homecare companies are there to serve you and you will be billed for services that you agree on.  You are not taking up their time and it only makes sense to want to know what services you will or will not receive.  There are no dumb questions.  Homecare companies seldom work with people that have been through this before and they know you don't fully understand their services.  People that are in homecare have to be very approachable.  You may have other questions, but be sure to ask the following questions.

Be sure to check with your local BBB (Better Business Bureau) and any licensing agency that might be available or appropriate.  While the vast majority of homecare companies are careful about who they hire, you don't want to hire the one company that has had problems with hiring good or honest people.

Set up a home interview time with your elder parent and the person from the homecare company to meet.  It won't be a long one but will help to break the ice a bit on the first day, especially if you are not there to get them started. 
Show the caregiver around the house where cleaning and other supplies may be located.  Talk over any questions that the caregiver has as well as those of the elder parent.  It is important that everyone feels comfortable in this setting.
Be sure to leave the caregiver with your contact information during the day in case additional questions need to be answered.  This is the time for everyone to come to a complete understanding of what is to be accomplished and get off on a great footing.

There are occasionally some strange dynamics in homecare situations.  While most are easily remedied, every once in awhile, someone will contract for minimal services and then expect that the caregiver will be only too happy to do more, without notice and without additional cost.

This is not a big problem for most people, but it should be pointed out that while your caregiver is there to help you, mutual courtesy and respect are a must.  Doing so will help all parties understand where the boundaries are and promote a terrific relationship.