Storing and parsing data with Arduino
Ideally, you would use a proper recursive JSON parser, like the one
suggested in prakharsingh95's answer. However, if code speed and size
are critical considerations, and if the format of the response is
narrowly constrained, then you can build a fast special-purpose parser.

Here is a simple example. It assumes the response has been stored in the
response char array:

static const char key[] = "\"action_status\"";
const char *p, *q;
char action_status[16];

for (p = response; ; p = q) {
    p = strstr(p, key);    // find key
    if (!p) break;         // not found
    p = strchr(p, ':');    // find ':' between key and value
    p = strchr(p, '"');    // find opening quotes for the value
    q = strchr(p+1, '"');  // find closing quotes

    // Report finding.
    strncpy(action_status, p+1, q-p-1);
    action_status[q-p-1] = '\0';
    Serial.print("Found action status: ");
    Serial.println(action_status);
}


Of course, in production code you would check that every strchr call
returns a non-NULL pointer. And you would check that the status you
found fits in your buffer.

Edit about the pros and cons of this method:

Compared to a proper JSON parser, this approach will give you smaller
and faster code, the drawback is it being more fragile. For example,
this will not detect a corrupted JSON, and it will return garbage if the
action_status happens to be a number instead of a string, or is a
string with embedded double quotes, or if the string "action_status"
appears as a value of some other field... A full JSON parser should
correctly handle all these cases.

On a non-embedded computer, a proper JSON parser would probably be the
only appropriate answer. On a constrained microcontroller environment,
you sometimes have to make trade-offs. In this particular instance, the
“best” answer depends on how much you weight speed and size vs.
robustness against unexpected input.