Good point there with 打 - typically when we say "to carry an umbrella", it's 带 (which means "to carry"). In this case, 打also means "to carry", but it typically doesn't mean that. 

Sometimes, if I get stuck on verbs that are being used in an slightly odd way, I'll do a google image search for the whole phrase. In this instance, 打着雨伞, and see what comes up. Hope that helps!