I have two squid proxies. First one have two network cards where one has global IP address and on other network card, there is private ip address, i.e 192.168.1.1. Here internet browsing is fine. I also enabled routing at this machine.

In my second squid proxy, there is only one network card where only one private IP address, i.e 192.168.1.10 alongwith default gateway of private ip address of first one proxy (i.e 192.168.1.1) is configured. I enable following options in squid.conf file as:-

cache-peer First-SQuid-Proxy parent 8080 proxy-only default

hierarchy-stoplist cgi-bin?

Due to above two lines and routing enabled at first one proxy, my clients also get internet access through second proxy, but when I stop routing at first one machine, then user could not get internet browsing from second squid proxy.

What is in this issue? Can we configured second squid proxy to get internet without routing enabled?

You need direct internet access for second proxy. That cache-peer is to share proxied objects, but you should add separate router/gateway, connect both proxies in parallel, and add switch between them and clients.