About once a month, the person in charge of our district's visual identity gets a request for one or more school logos. Sometimes, the PTA wants to use the logo on a newsletter or printed piece. Other times, it's a company looking to make t-shirts or something else they can sell to students. We get enough requests like this that our graphic designer has the process down to a science: figure out if it's reasonable, make sure the principal is aware and approves, then figure out what computer file format will work best.
The thing is, I can't remember a time when someone requested the school district's logo.
That's the norm, though. There are tons of businesses that "adopt" the schools near their brick-and-mortar locations, but many fewer reach out to help the entire district. When we post on Facebook about individual schools, we get excited replies from parents, but district-wide stories just get "likes" and comments that read like form letters.
People don't connect with school districts; they connect with schools. We care about the school we go to, but it's hard to care about the district because it's not familiar and it involves so many unknowns. Parents haven't interacted with the district office (or even those other schools) every day for years, the way they did when their kids went to the neighborhood school. Worse still, bad news travels faster than good, so sometimes that's all that parents have heard about the district.
District-community connections do exist, though, and it's not impossible to strengthen them. We post tons of news from all our schools, for example, so people get the idea that there are always good things happening at every school. The events that logistically have to happen at the district-level help bridge some of the gaps. And, some things are just easier to go ahead and take care of at the district level.
...but don't spend all your time working on the district website. Take a step back from rigid district-wide templates and let schools develop their own styles and personalities. Even better, look at how the schools work with their community now, and help them build their websites around those ideas. Definitely embrace their cultural touchstones; post pictures of beloved long-time staff members and play up their traditions.
If you can redirect resources toward helping schools strengthen their own community connections, you'll get a lot more bang for your buck. When your district needs support, you'll still be able to make that appeal to the community, but you'll do it through the schools. People will line up to help the district when the teachers they've known for years point out that their own neighborhood school will see the benefit.