Q:

How to read “A = (πr)²” so as not to mistake it for “A = πr²”

None of the 26 answers given here, or the 5 answers given here mentions any similarity between the pronunciation of  E = mc² and  A = πr², yet I still remain confusioned as to what distinguishes the reading of E = (mc)² and A = (πr)² with the reading of the first two.

A:

Most of the mathematicians I know would pronounce these as:
A = πr²
A equals pi r squared
A = (πr)²
A equals pi r all squared
There is no fixed rule, so this really comes under the category of primarily opinion based.

A:

To read A=(πr)2 to a class, items in parentheses tend be categorized as the quantity with contents terminated by a pause in speech.
Per the Oregon State math read-aloud accommodation

Parentheses - read as “the 
  quantity”
  3(x + 2) is read as “three times the quantity ‘x’ plus two”
  (y – 5) ÷ 6 is read as “ the quantity ‘y’ minus five (pause) divided by six”

Which is to be interpreted for the question as:

A equals the quantity pi 'r' (pause) squared

Edit to add: "Is it pi 'r' or is it pi times r?"

Expressions containing variables (any letter may be used as a variable):
  “‘V’ equals four thirds pi ‘r’ cubed”

I think it should be noted that vocalization of a math(s) expression is likely to be accompanied by a written version of what's being said: either on the board and/or for students to transcribe. In the case of the AmE the quantity, this signals the beginning of a parenthetical expression (i.e., write or read an opening parenthesis) where the pause signals the end (close parenthesis). It may be trivial to retroactively assign parentheses as the expression is complete (all squared), so this distinction is not as problematic as it may seem, especially since it can be argued that "all" is an unambiguous demarcation point between the parenthetical expression and the accompanying exponent, versus a pause in speech.
Reviewing the body of the question:

.. the pronunciation of E = mc² and A = πr², yet I still remain confusioned as to what distinguishes the reading of E = (mc)² and A = (πr)² 

Taking the latter first:

E equals the quantity of mc (pause) squared

is read the same as 

A equals the quantity of pi r (pause) squared.  

Contrast this with: 

E equals m c squared, 

or 

A equals pi r squared 

(no pause, and without the quantity) one understands that the exponent only applies to the preceding variable/operand. If one asks how the written version distinguishes the coverage of exponent to its applicable operand (e.g., why the exponent applies to the entire expression in parentheses), that is indeed a math(s) question. 
Further edit:
There appears to be some confusion that the quantity is dependent on the content within parentheses being a sum or difference rather than being agnostic to the contents. It can be argued that a quantity can contain any expression but while it has been generally associated with add/subtract, it's the parentheses that determines whether the quantity is being used or not.
Additional usage examples of the quantity:
mathforum post
http://www.mastermathmentor.com/mmm/ReadingMath.ashx (How to read to the blind and dislexic). Replaces the pause with "end quantity". 
Yes, while the expressions tend to be sum/difference, every discussion talks about the parentheses being addressed, not the content, and at no time is there any distinction being made about the expression within parentheses changing the attribute describing parentheses. 
Further, quantity itself does not distinguish the expression configuration as does sum, difference, or product. The free dictionary does not indicate anything about how an expression is defined for the definition of quantity. 

A:

A = (πr)² can be read as "πr all squared", and is another way of writing π²r² which is also the same as π² x r² (pi squared times r squared).  In other words, the original formula [A = (πr)²] is just a different, and sometimes shorter way of writing the equivalent formulas.  It doesn't save much time in this particular example, but it would if you wrote A = (a + b)² instead of the equivalent A = a² + 2ab + b²

