How can I delete old messages on my iPhone to free up space (NOT the whole message thread)
You can do exactly what you want as follows:


Go to Settings > General > iPhone Storage
Wait a little while as the screen starts populating with data for each of your apps
Within the list that appears, tap on the Messages icon
Now you’ll be presented with a number of options for saving storage space 
Select one or more of those options (see below)


Options 


Enabling Auto Delete Old Conversations will automatically delete all messages (and their content, incl. photos etc) that were sent or received over 12 months ago
Tapping on Review Large Attachments will allow you to view all attachments. You can also swipe left to delete them. Likewise, you can tap on the Edit option at top left of screen to select multiple attachments to delete.
You can actually choose to delete individual items by reviewing them within the Documents &amp; Data section at the bottom of the screen. This allows you to review Conversations, Photos, Videos, GIFs, Stickers and other items (e.g. tapping on Photos will allow you to review and/or delete each one individually).