How long does the virus live on the pillowcase?
No virus would transfer to the pillowcase, so zero risk.
The only practical transmission route is kissing your child with an active cold sore. Hence keep these away from the baby and this incredibly reduces transmission chances.
Although people do not intentionally infect their children, it happens a lot. If it isn't the parents, then it is other family, fellow classmates or eventually partners. This is not desirable, but nor is it to be feared. People have an amazing immune system that deals with the virus incredibly well.