Have you been able to do this previously and this problem has recently appeared?
The 403 error suggests there is a permissions problem, the source of which may be some server security settings or a security-related plugin.
As an initial test I would de-activate all non-Toolset plugins and then try again to update your parametric search.
If that works it would appear to be a conflict with another plugin, and you can narrow down which one by gradually re-activating your other plugins, either singularly or in batches, re-testing each time, until the problem re-appears.
In which case, if we can rule out conflicts from your plugins or custom theme code then it must be an issue with your server settings.
We don't experience the problem ourselves and it isn't reported by other users.
I would consult with your systems admin or your hosting provider about why admin-ajax.php is being restricted, or you could consider adding a rule to your .htaccess file to allow access.
It's very strange, why does it work perfectly for certain things (like text and post search box) and not for others? It's the same POST request to the same script so it would strongly surprise me if it has something to do with server settings. Otherwise it should never work I think.
What happens in the background when saving a view filter template-content? Does Toolset "analyse" is somehow? And is there a case where it return a 403?
I suggest you clone your site to your local computer and test it when running on localhost.
If that is not possible I can request a test installation on one of our servers where you can upload your site to see if the problem can be reproduced.
Given that we have no other reports of this issue I'm almost certain it won't be reproduced on a different server, but if you test locally and find that it is then please let me know so that I can pursue this further.
My host tells me there are a lot of by the plugin on submission. For protection IP's get blocked after x requests in a short time. Could that be the cause?
Before I start pulling it all local: are there more (sub)requests happening when submitting the field with filter-code in it than just with text?
If yes, then it will probably be the cause. If not, it needs more digging.
I think perhaps I should get a copy of your site so I can do some testing.
Could you please use one of the following two free plugins to create a snapshot of your site and upload it to a public url (such as dropbox or google drive) so that I can reproduce a copy of your site locally to test. You may want to set up a temporary admin account for me before you do which you can later delete.
* Duplicator. You will need to create and upload both the installer and archive packages as shown in the demo video on their plugin page: https://wordpress.org/plugins/duplicator/. You can omit un-required files such as .log, .zip files etc.
* All-in-One WP Migration. This is simple to use and is less prone to fail because of server settings than Duplicator. https://wordpress.org/plugins/all-in-one-wp-migration/. But note I can only import archives up to 512Mb in size. If they are not relevant to your issue, you can omit media files which will reduce your archive size.
I will mark your next reply as private so that you can give me the url to retrieve the backup files.
Can you also confirm the url of where I can see the problem.