Escrow is a neutral third party that carries out the agreed upon terms set out in the fully executed (signed by buyer and seller) purchase agreement. Escrow has nothing to do with inspections or obtaining financing.
You might feel more secure with the process if you and your parents interviewed some agents to walk you through the procedural stuff. You can hire separate agents or one agent to represent you and your parents.
If you decide you'd like the assistance of a Realtor, you might ask if they will agree to help you for an agreed upon fee, as your transaction doesn't involve some of the steps that other transactions require.
You might want to look into Prop 58, which speaks to the potential tax benefits with regard to parent-child property transfers. Here's the link: http://ocgov.com/gov/assessor/realproperty/transfer/
Best of luck! Let me know if you have any other questions or would like my assistance.