When creating a new project today, am I allowed to license my software using the older version of licenses such as GPLv2 (vs. GPLv3), or BSD 4 clause (vs BSD 3 clause).
Am I only allowed to use the latest version of these licenses instead?
You can use any version of any license you please. You are the author. You do what you want.
No FOSS license could say something to the contrary, including no such license could restrict you to use other licenses or other version of a license for your original work .
Practically each new "version" of common FOSS licenses is in itself a new license. Most "versioned" licenses have terms that may allow to use a newer version of a license for code using a previous version, but not always. This is a permission in all cases, never a restriction.
In some cases such as the GPL family the relationships between all these versions can start being .... it's complicated!.
If by new you mean new from scratch you are free to choose any license (or "version" of a license) that suits you. But try not to think of it as versions but rather just different licenses. For example, the GPLv3 and the GPLv2 are not even compatible with each other and are quite different from several aspects. The GPLv2 is still a logical choice for many projects.
As some sort of rule of thumb I would suggest that if a license is NOT found among the OSI approved licenses you need to have a very specific reason to use it for your project.
If you are creating a new project based on somebody else's work you need to choose the same or a compatible license for your project. For BSD and other similar non-copyleft licenses this is normally no issue as they can mingle with each other happily, but for reciprocal licenses like the GPL a single contradictory term between two licenses means that they cannot be mixed.