You can consider it as a verb かわいがる + て +　いました。 かわいがる means to love, take care, pamper and fondle some animal/someone. It's originated from adjective かわいい as explained above, but it has nothing to do with the verb 飼います。

Agree with the answer above. Here is the additonal explanation, がっています in かわいがっています, it comes from the word/stem of かわいい　（kawaii）. This infliction shows what 3rd person may think about something. We use "adj i+gatte imasu" rather than using the simple form, kawaii, because we don't know exactly what third person think about something. This form shows us the subject, whether the first person or the third person. 二人は子どもがいなかったので、シロというイヌをとてもかわいがっていました。 Because they don't have children, they really love shiro, their dog

This is a form of the adjective かわいい。And がる is a suffix that sticks to the stem and means "to look like, to seem" or "to pretend". It is used with adjectives and, as I feel, is more about the feelings and impression of the watching one. Little example: 女の子はいつも寂しがる。They look 寂しい (lonely). So all together: As they (we) had no children (or As there were no children), they (we) thought that the dog Shiro was very cute. And an article from superhelpful dictionary: http://dictionary.goo.ne.jp/leaf/jn2/46427/m0u/%E3%81%8C%E3%82%8B/