No, if a message is sent through TalkingParents, the receiver cannot preview the message. The message must be opened to be viewed, and this action will prompt a timestamp.
For an example of this, please take a look at the screenshots below:
1) When you receive a new message from your co-parent, the messaging subject should display
New message from (your co-parent).
2) You will need to open this subject to view this message. This action leaves a timestamp.
3) Once all messages are viewed within a subject, you will then see a preview of the most recent message.