What is the difference between parent.frame() and parent.env() in R; how do they differ in call by reference?

It would be helpful if someone can illustrate this with a simple example?

Also, where would it be useful to use parent.frame() instead of parent.env() and vice versa.

parent.env is the environment in which a closure (e.g., function) is defined. parent.frame is the environment from which the closure was invoked.

f = function() 
     c(f=environment(), defined_in=parent.env(environment()),  
       called_from=parent.frame())
g = function() 
     c(g=environment(), f())

and then

> g()
$g
<environment: 0x14060e8>

$f
<environment: 0x1405f28>

$defined_in
<environment: R_GlobalEnv>

$called_from
<environment: 0x14060e8>

I'm not sure when a mere mortal would ever really want to use them, but the concepts are useful in understanding lexical scope here

> f = function() x
> g = function() { x = 2; f() }
> h = function() { x = 3; function() x }
> x = 1
> f()
[1] 1
> g()
[1] 1
> h()()
[1] 3

or in the enigmatic 'bank account' example in the Introduction to R. The first paragraph of the Details section of ?parent.frame might clarify things.

Environments are pervasive in R, e.g., the search() path is (approximately) environments chained together in a sibling -> parent relationship. One sometimes sees env = new.env(parent=emptyenv()) to circumvent symbol look-up -- normally env[["x"]] would look first in env, and then in env's parent if not found. Likewise, <<- looks for assignment starting in the parent.env. The relatively new reference class implementation in R relies on these ideas to define an instance-specific environment in which symbols (instance fields and methods) can be found.