A rail that moves under a hand is worse than no rail, because Dad will trust it right up until it pulls him off the stairs. The question is not whether the rail looks solid. It is whether each anchor goes into structure that will hold a grown man leaning his full weight on it.
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Find out what is behind the wall
Start with the wall, not the rail. Use a stud finder, then confirm with a small test hole where the fastener will go, because a finder can read a pipe or a wire as wood. Plaster over lath sounds different from drywall and holds screws differently, and a rail anchored only into lath will eventually pull out. If the first screw lands in a hollow, stop. You are looking for solid wood, or blocking someone built between the studs, reached through the wall framing or the stair stringer.
Decide whether you mount it or hire it out
Mounting it yourself is reasonable when three things are true: you can find framing at every anchor point, the existing rail or post is fixed to something solid, and the job does not require opening the wall or cutting into the stair structure. You also need a drill with enough power for lag fasteners and a second person to hold the rail while you set it. If any of those is missing, the job has moved from a weekend fix into carpentry.
Hire it out when the wall is hollow where the rail has to go, when the stringer is cracked or loose, when the rail needs a new post or a return at the top or bottom, or when the fix means patching and repainting afterward. A carpenter or finish contractor can also tell you whether the work needs a permit, and your local building department can answer that for your address. Do not assume the answer is no.
What to ask the contractor before you sign
Ask how they will find framing and what they will do if none is there. The answer tells you whether they plan to fasten into structure or into drywall anchors. Ask whether the quote includes patching and repainting, whether they remove and reuse the old brackets, and what fastener they will use. Ask who carries their insurance and get a certificate. Put the scope in writing: number of anchor points, rail length, end returns, and finish. A quote given without someone looking at the stair is a guess.
Test it the way Dad will use it
Have Dad grip the rail at the spot where he holds it going down, and let him lean, not you. Watch the anchor, not the rail. If you see movement at a screw head or feel the wall shift, the rail is not finished. Check it again after the first few weeks, since fasteners can settle in a wall that moves with the house. A rail that passes is useful. A rail that moves is a trap with a handle on it.
Your next move tonight
Go to the top of the stairs, run the stud finder along the wall where his hand goes, and mark every hit with painter's tape. If the marks line up with framing at each planned anchor, mount it this weekend. If they do not, call a carpenter and ask the questions above before anyone opens the wall. For names of local trades and services, start with Local Help in St. Louis.