Installed new SSD, but I have no Snow Leopard disk. Can I install Lion from Internet?
Lion / Mountain Lion's recovery is only present on hard drives that have the system already installed.  Internet Recovery (likely what was used with your Dad's Mac) is only built in to newer systems that shipped with Lion or Mountain Lion. Apple support page documenting this

In your situation I think your best bet would be to download Mountain Lion using your MacBook Air, build a Mountain Lion installer disk using the application below, boot to the install disk/drive by holding option while powering on your Mac and selecting the disk and installing as usual from there.

Lion Disk Maker

Hope that helps!