Joanna,
Thank you for sending over your question to the Community ToolBox. There are several things to think about when you say increase family engagement. Who are the families? Do they represent groups who are learning a new language as their second language, groups who have been traditionally marginalized, etc? I would suggest you consider some of these questions to think through some ways that you can provide resources to the families. What are their needs and what value will they obtain if they were to become more engaged in your programming? Also, you may need to bring in someone who represents their community and someone they deem as an active member in their community and work through them to invite families. These are some initial questions and suggestions for you to consider.
The Community ToolBox also has a resource to help guide you and to increase participation and engagement across groups: https://ctb.ku.edu/en/increasing-participation-and-membership.
I hope this helps you and thank you.