I did the connection a month ago, albeit from SK to AA. But SK also arrives at T1 so the procedure should be the same.
Follow the signs for Transfers when arriving. In order to get from T1 to T3 you (or your parents) will need to take a bus. They will call the bus from the transfer desk and it should arrive within 6 minutes. They drive you to T3 where you will pass through security. (I doubt that you can walk.)
If you do not have a boarding pass (I didn't) it was enough to show the itinerary from AA. After security a gentleman showed me to the AA gate where AA printed boarding passes, looked thorugh my passport, asked if I packed my bagfs myself, asked why I was going to the US etc.
I hade 1h15min to my disposal and it worked out fine - but the SK flight was on time (actually a bit early) and noone else was transferring so every thing went very smooth.
If AY is delayed you may get in trouble as they most likely wait for connecting passengers. But there are several possibilities to connect via LHR.