Making a <br> node start a new paragraph isn’t something that the DOMParser can currently do—you can of course make it insert any node in its place, for example a hard_break node, as I guess you found out, but there’s no way for a parse rule to force the current parent node to be closed.
That might be a good thing to add to the library, but I’m not sure when I’ll have time to work on that.
See the test in the patch I linked. You can’t add such a rule to a schema node spec, since it’s not creating any nodes. You have to manually create your DOMParser to include the rule.