You can check if you can reconfigure your router and tell it not to respond to ping.
As you have seen in this thread, some people thing router firewall is adequate and others think it is not and each computer within the local area network need its own software based firewall. Which side you will be on depends on you.
FIrewalls opens and closes ports and you will have to be more specific on what network resource you want other computers on your network to have access to. You have to be specific on what program for example doesnt work correctly when the firewall is up for people to know which port need to be open