<?php

$json_string = '[
    {
        "department": "CIS",
        "name": "Intro to CIS",
        "sections": {
            "01": {
                "regNum": "012345",
                "days": "MWF",
                "startTime": "900",
                "endTime": "1030",
                "labDay": "M",
                "labStartTime": "1300",
                "labEndTime": "1500"
            },
            "02": {
                "regNum": "098304",
                "days": "TR",
                "startTime": "1300",
                "endTime": "1500",
                "labDay": "",
                "labStartTime": "",
                "labEndTime": ""
            }
        }
    },
    {
        "department": "MATH",
        "name": "Intro to MATH",
        "sections": {
            "01": {
                "regNum": "012345",
                "days": "MWF",
                "startTime": "900",
                "endTime": "1030",
                "labDay": "M",
                "labStartTime": "1300",
                "labEndTime": "1500"
            },
            "02": {
                "regNum": "098304",
                "days": "TR",
                "startTime": "1300",
                "endTime": "1500",
                "labDay": "",
                "labStartTime": "",
                "labEndTime": ""
            }
        }
    }
]';

$departments = json_decode($json_string, true);
$result = array();
foreach ($departments as $dept) {
    $name = $dept['department'];
    foreach ($dept['sections'] as $sectno => $secdata) {
        $secdata['dept'] = $name;
        $secdata['section'] = $sectno;
        $result[] = $secdata;
    }
}

var_dump($result);