who signs the timesheets?

I get what you mean but your time slip has nothing to do with giving report.

The client or family member's signature on the time slip is their agreement that you worked the times listed.

One of the financial forms they probably signed at the start of care was their agreement that their signature on the time slip is an agreement to pay for the hours.

At my agency the time slip has to be signed by the client or a family member not another employee unless there is no other option.

If the parents are not there, the oncoming nurse may sign the sheet or you can write "patient unable to sign", if for some reason you are not leaving the patient in care of another (there are some exceptions when this might be the case). If the oncoming nurse is late, you wait, unless of course, the family is there. Then you report to the family and leave the patient in care of the family member. If the agency requires the family to sign the sheet and they are not there to sign it, you have an issue for the Labor Board. You may never return to the home, so then, how are you to be paid? When you leave a shift, there is to be no unfinished business, including your time sheet.

They are telling me to go drive to the pt home and have the parents sign it.

Not necessary and not your responsibility. Take it to the Labor Board.

On some timesheets it says "reported off to and left in care" below the signature line.which is confusing.

This only explains who you gave report to and turned over the patient to. It makes it clear for legal purposes. Oncoming nurse, family member, or whoever, as the case may be. But that line does not take the place of the representative signature. Unless, they aren't there to sign it, then it is the oncoming nurse, "patient unable to sign", or "so and so not available to sign" or "whatever the case may be". This agency is messing with you.