Search for a certain combination of strings in serial and read the following characters until another set character stops the search

Firstly you should really be working with a C string (char *) instead of a String object, otherwise you will be massively fragmenting your heap with the string manipulations to come.

Once you have your data in a C string (char array) you can use strtok_r to split the string on the | character, then for each substring further split it on the :. You need the _r variant of strtok to make it "re-entrant" so you can have two tokenizations at once nested.

For example (untested):

char *data = "<Idle|MPos:158.500,0.000,0.000|FS:0,0|Pn:H|Ov:100,100,100|SD:0.21,/instruction.nc>";
char *pair; // Pointer to a K/V pair
char *pairptr = data; // Pointer for outer strtok

// Split the string on the first |
pair = strtok_r(data, "|", &pairptr);

// While we have a K/V pair
while (pair) {
    char *kvptr = pair;
    // Split the substring into two parts on the colon
    char *key = strtok_r(pair, ":", &kvptr);
    char *val = strtok_r(NULL, ":", &kvptr);
    if (val) { // If we split the data correctly:
        if (strcmp(key, "SD") == 0) { // If the key is "SD" then
            Serial.print("SD data: ");
            Serial.println(val);
        }
    }
    // Get the next string portion
    pair = strtok_r(NULL, "|", &pairptr);
}

Once you have your val you can further split that down on commas in the same way as it's been split on the colon.