Read one value from continuous data stream at a lower frequency
You're thinking backwards. You don't want to read slower (which is what delay does), you want to only send every 10th entry.

That is, count the lines as they come in. When you hit 10 you print one and reset your count.

In other words:

static int count = 0;

if(newParoData == true){

  count ++;
  if (count == 10) {
      count = 0;

      // (Why copy the line just to send the copy...?)
      for(int i=0; i&lt;16; i++){
        paroToSend[i] = paroChars[i];
      }
      Serial.print(paroToSend); Serial.print('\n');     //transmit Paro data with a new line for each value
  }
  newParoData = false;
}