Why do we still use the 'P' subscript in a thermodynamic derivative such as (dH/dT)P?
In thermodynamics, the number of degrees of freedom available to describe a system is given by Gibbs' phase rule:

$$
F = C - P + 2
$$

Here, $C$ is the number of components in the system, $P$ is the number of phases in the system, and $F$ is the number of resulting degrees of freedom.

Consider a pure substance in a closed system.  One can name a wide variety of thermodynamic parameters to describe its state: temperature, pressure, specific volume, internal energy, enthalpy, entropy, etc. However, Gibbs' phase rule says that we can only independently specify a subset of these parameters before we run out of degrees of freedom:

$$
F = 1 - 1 + 2 = 2
$$

So, in any mathematical description of (in your example) the enthalpy, $H$, it can only properly be a function of two independent variables:

$$
H = H\left(X_1, X_2\right)
$$

But: which two variables?

Simply writing the partial derivative $\partial H \over \partial T$ indicates that temperature is to be one of the two available degrees of freedom:

$$
H = H\left(T, X_2\right)
$$

Now, though – what is $X_2$? Well, including the subscript on the differential is a way of explicitly indicating what that other independent variable is:

$$
\left({\partial H \over \partial T}\right)_P \quad \Longrightarrow 
\quad H = H\left(T, P\right)
$$

As porphyrin rightly notes in a comment, this applies to multivariate calculus generally, not just the calculus of thermodynamics.