In the long-run I would recommend upgrading PHP versions, but I understand that this can be a lot of work. There are many good guides on upgrading versions. I recommend going through the version upgrades one-by-one, dealing with deprecated features and other potential problems one-at-a-time and testing each script.
In the short-term, I'm not sure if this would work, but PDO is supported in 5.2 (actually in 5.1 even) and as it simply passes info to the PostgreSQL library, I would expect SSL support to work. I have used it in 5.4, and I am not aware of any changes made between those two versions.
There is a solution here, by IMSoP, that explains how to get this working and points you to more official documentation if you need more info: Connecting to PGSQL over SSL via PHP PDO
If this fails, a second approach that I would try would be to create an SSH tunnel to connect to a local port. An example would be:
ssh -L 127.0.0.1:5443:127.0.0.1:5432 username@remoteboxaddress
Then you can initiate what looks to PHP like a local connection on port 5433
, but it will actually be a (secure) SSH tunnel to the remote box. This will also require that the remote box is configured to accept (non-encrypted) local connections, and that you have SSH access to it. Then, if you test it out and it works, you can make a script to start the tunnel on boot, or better yet, when the webserver is started; this necessitates using an authentication method like keys stored in a file, that doesn't require the entering of a password. So...there are a lot of "if's" here, in terms of whether or not you will be able to get this solution to work, but it can be done if you have root access on both servers, and occasionally if you do not, depending on how they are configured.
Hopefully, one of these two solutions will work for you.