In my opinion, building a Viral Product takes more than just Marketing or Luck. I believe it begins with an understanding of your customer, and their desire to Share information within the Context of your Product. Taking this into consideration when designing the Product (or it's features) will allow you to build a Product that provides an environment that facilitates viral growth. Weather or not the Product actually goes viral will depend on how useful / valuable the Product is, in conjunction with your success rate of understanding your customer, and integrating this understanding into the Product.
In your example, I would start with some customer validation. Are you currently talking to your customers? If not, then this is probably an area of concern as you are likely out of touch with what they actually want. If you are talking to your customers, then you would want to modify your interaction with them to include some questions about what drives them to share, and what types of information they like to share, as well as who they would want to share it with. Then you basically take what you've learned and integrate it into your Product.
Going off the top of my head (this has not been validated, just my own understanding from knowing lots of parents), I would key in on the fact that Parents like to Brag about their Kids. So when their Children accomplish something cool, they want to tell all their friends about it. I would integrate this into the Product by identifying common Accomplishments that children experience in life (just learned how to walk, just learned how to talk, learned to ride a bike, etc,etc all the way up to graduated from college, got a good job, bought a house, and/or started a family). By identifying these moments / memories as Accomplishments (and perhaps assigning them a special badge of some sort), you will increase their value in terms of something worth sharing. Your Product already allows Parents to share specific moments / memories, but because these moments / memories are special, you will increase the Customers (Parents) incentive to share them, which becomes a vertical for potential viral growth. By following this train of thought (which always starts with customer validation) you can identify multiple verticals for potential viral growth. Essentially, you just want to learn and understand what makes your Customers share information, and then facilitate the sharing of that information. Good luck!