Apple - Is it possible to have my iPhone 5 on silent except for one specific person?

This is possible using Do Not Disturb:

1. Open Settings.app
2. Go to Notifications > Do Not Disturb
3. Select "Allow Calls From"
4. You have several options. Either create a group, or have only those few people in your Favorites. The best is probably to create a group as that allows you better flexibility. After you have created the group with only the people that you want to allow to reach you, select it from the list.
5. Now, to enable silent mode except for those two people turn on Do Not Disturb in Settings.app.

Creating/Editing Groups: This isn't as easy as it appears on iOS. See my question about it. As mentioned in the accepted answer, your best strategy for creating or editing groups is to go to iCloud.com on your computer. Quoting the answer:

"No, there is no way to create or edit contact groups on iOS. I share your frustration! The best workaround is to use iCloud.com (or your primary Mac, if you have it with you)."

But you can add contacts to a particular group, (though it isn't straight-forward):

1. In Contacts, tap the Groups button in the top left
2. Tap "Hide All Contacts"
3. Tap the group that you want to add the contact to (making it the only visible group)
4. Tap Done in the top right
5. Tap the Add button (+) in the top right

So, the easiest is to just use a Mac (if you have one) or iCloud.com to create and add people to a group. After this is done, the new group with the people that you want to reach you should appear on iPhone.

There is a feature in the newer versions of iOS that is called "Emergency bypass". You can find it in Contacts > [person] > Edit > Ringtone > Emergency bypass.

Your iPhone will ring regardless of Silent or Do Not Disturb modes on or off.