Free variables, parameters, dummy names -- what policy to choose? A long lead-up (sorry!) to a short question. Start with a semi-formal argument in Loglish:

(1) $\quad$ for every $x$, $x$ is $F$
(2) $\quad$ for every $x$, ($x$ is $F$ $\to$ $x$ is $G$)
Arbitrarily choose some object in the domain, a: then we have
(3) $\quad a$ is $F$
(4) $\quad a$ is $F$ $\to$ $a$ is $G$
(5) $\quad a$ is $G$
But a was arbitrarily chosen; so
(6) $\quad$ for every $x$, $x$ is $G$

Obviously, something we have to explain to beginners is: What is '$a$' doing here? This plainly isn’t an ordinary proper name, attached once-and-for all to a determinately given individual. On the other hand, it isn’t an anaphoric pronoun either. It is a sort of ‘dummy name’, or ‘temporary name’, or ‘ambiguous name’, or ‘arbitrary name’ – perhaps none of those labels is entirely happy, though all are in use. Let's say it is a ‘parameter’.
When we go fully formal wanting to regiment such arguments into an official first-order language, we need some symbols to play the role of parameters, as informally played by '$a$'. There seem to be three policies on the market, used by different textbooks:

(A) We can use the same symbols both as variables-tied-to-quantifiers and as stand-alone parameters (in conventional jargon, the same symbols can appear as both ‘bound variables’ and ‘free variables’).
(B) We can use the same symbols both as proper names and as parameters/dummy names (with just some of these symbols getting a fixed denotation in a given language).
(C) We can use distinctive symbols for proper names, for bound variables, and for  parameters/dummy names.

Each policy can be made to work. Policy (A) has historically been the most common one among logicians. And it does conform best to the practice of mathematicians who casually move between using letters as dummy names (‘Suppose $m$ is the number of positive roots . . . ’) and using letters of the same kind as quantified variables (‘For any integer $n, (n + 1)(n − 1) = n^2 − 1$’), letting context make it clear what the symbols are doing. However, when we go formal, overloading symbols like this does cause complications: we have to spell out very careful rules for their double use if we are to avoid getting into tangles.
Policy (B) is easier to work with, but hasn’t anything special to recommend it. Once we have distinguished, at the semantic level, the fixed-denotation names built into a language from its further supply of dummy names, why not highlight the distinction by using syntactically different symbols for the two styles of name?
Policy (C), then, may be less economical but it does make it easier to keep track of what is going on. Variables only appear bound, parameters only appear free. It conforms to the good Fregean principle of marking important differences of semantic role by using different styles of symbols. It has distinguished antecedents, e.g. in  Gerhard Gentzen. It's the policy in e.g. Richmond Thomason's impressive textbook.
Which is the easiest policy for the beginning student to understand? I'd say (C). But -- and here at last is the question --

Is there any strong reason -- issues of ease of understanding apart -- for preferring the more common policy (A) to the less common policy (C)?
The only thing I can think of is that by adopting policy (A), which makes it ok to use free variables in the statements, we can use it more naturally as a system that can prove properties and relations of formulas in general. For example, you can prove that $P(x)$ and $\neg \neg P(x)$ are equivalent wff's.

But you obviously you can only do those kinds of things if you have to be really careful about how the variables are used, i.e. whether it's one of those 'temporary' variables or not. And certainly for beginning students we just work with sentences, rather than formulas in general. So, policy (C) has a lot be said for it, I agree.
Not sure what you mean by a &quot;parameter&quot; here. Assuming a &quot;dummy&quot; variable is the same as a bound variable.

Start with a semi-formal argument in Loglish:
(1) $\quad$ for every $x$, $x$ is $F$

This is an assumption. $x$ is a bound (dummy?) variable here.

(2) $\quad$ for every $x$, ($x$ is $F$ $\to$ $x$ is $G$)

This is an assumption. $x$ is a bound (dummy?) variable.

Arbitrarily choose some object in the domain, a: then we have
(3) $\quad a$ is $F$

This is an assumption. $a$ is a free variable.

(4) $\quad a$ is $F$ $\to$ $a$ is $G$

Applied Universal Specification here.

(5) $\quad a$ is $G$

Applied Detachment (MP) here

But a was arbitrarily chosen; so
(6) $\quad$ for every $x$, $x$ is $G$

Applied Universal Generalization here. $x$ is a bound (dummy?) variable.