Your own messages show Delivered once they're in the record, and Read with a time once your co-parent has opened the thread and seen them.
Read means the message was on their screen. It doesn't mean they took it in, agreed, or are able to act on it yet. Treating a read receipt as an answer is a reliable way to start an argument.
You can't turn read receipts off for yourself only — they'd stop meaning anything if either parent could.