Most of the trouble between work and the parent calendar is not that the job says no. It is that nobody has agreed to a specific, written arrangement, so every afternoon call gets handled as an emergency. Before you ask for anything, compare the three kinds of schedule change that most workplaces can offer, and check which one covers the hours you actually lose.
Local option: New Plan Care gives families in the St. Louis area in-home help so a parent can stay home: companionship, help with bathing and dressing, meals, and overnight care. Call (314) 405-0887.
Start with the written policy
Find the document your employer already has: the handbook, the remote work or flexible schedule policy, or the time-off section. Note what it says about start and end times, remote days, and making up hours. Written policy is the floor you can point to. If the policy is silent or unclear, that is useful information too, because it tells you the arrangement will need to be negotiated and written down. For questions about your specific rights, a qualified employment advisor can answer what a general guide cannot.
Compare the three options
A shifted start moves your day earlier or later. It helps when the gap is the problem, but it does nothing for a midday call from Mom's house, so ask whether the shifted start leaves you free for the pickup window. A remote afternoon gives you a place to be when the phone rings, but only if you can take a call and step away without losing the thread of a meeting. Make-up hours let you leave early on a split-week day and finish later. That works well for some jobs and poorly for jobs with fixed shifts or client coverage.
For each option, ask four questions. Which hours does this cover, and do they match your real problem hours? What does the team need from you during core hours? Who covers for you when you are out? And is the approval written, with a check-in date you can revisit, or is it a verbal promise from one manager who may change roles?
Bring the comparison to your manager
Show up with one option, the one that covers your worst hours with the least strain on the team, and a plan for how the work gets covered. A short written summary is easier to approve than a long conversation. Name the trial period, the check-in date, and the signal you will both watch for. If the answer is no, ask which of the other two options might work and what would need to change for it to work.
A flexible setup is not a favor you have to keep justifying. It is a working arrangement with the same expectations as any other, and it holds up best when both sides can see exactly what it covers. If you want to think through the afternoon coverage side before that conversation, the Local Help in St. Louis page lists places to start.