If we consider a space spun by the thermodynamic variables $(p,V,T)$ (or some other work-like conjugated variables, like the magnetic moment $\mu$ and field $B$), then an equation of state may be given as the level set of a function (given by experimental data or some a priori considerations) defined on that space, i.e. $$F(p,V,T)=0$$ This level set viewpoint gives the explanation of why thermodynamical values are mutually dependant, i.e. why giving the value of two variables defines the third. Furthermore, that level set may be regarded as a 2-dimensional manifold $M$ embedded in the space of thermodynamic variables. Getting information about entropy or thermodynamic potentials may be viewed as a Legendre transform of that equation, but that's another story.
A (parametrised) thermodynamic path $\gamma(\lambda)\subset M$ can then be defined in the way you grasped in your question, as a curve on the 2-manifold $M$. In other words, let $\lambda\in\left[a,b\right]\subset \mathbb{R}$. Then we may define: $$\gamma:\left[a,b\right]\to M$$ $$\gamma(\lambda)=(p(\lambda),V(\lambda),T(\lambda))\in M$$ Note that one set of boundary values can define an infinite amount of curves.
For practical calculations in general physics we don't usually need to invoke so much rigour, so we just jumble around with differentials like apples until we get something satisfactory. In case of phase transitions, however, it becomes crucial to analyze the properties of discontinuities that arise in parts where the manifold is not smooth anymore. That, in part, is the work for which this year's Nobel in physics was given.
