Will only start with foot on brake

I have a 1999 Chevy Prizm with automatic transmission.
Recently began to act up....will only start if my foot is on the brake pedal. If the brake isn't pressed, I will only hear a click 3 times out of four attempts. It will occasionally start normally. The battery connections have been cleaned and are tight. Good voltage too.
I'm guessing that its some type of switch that's starting to go bad.
I'd appreciate any suggestions on what to test and/or replace. I have an Ohmeter ready.
Thanks in advance.
Tom B.

My guess is the starter. The brake pedal has nothing to do with it starting. Check to see if your getting 12V at both terminals at the starter with the key in the start position. If you don't have 12V at the solenoid terminal [smaller wire] with the key at START, suspect the neutral safety switch or starter relay. The starters go bad in these cars

If you don't have 12V at the solenoid terminal [smaller wire] with the key at START, suspect the neutral safety switch or starter relay.

Thanks for the reply, John.
I'm having difficulty seeing the wires at the starter. The intake manifold obscures access, even with an extension mirror. Is it best to get near the starter from the bottom of the car, or from the front/top if it needs to be removed?
Tom B.

From the bottom.

Jack, support, the car. Remove passengers side spash guard. The solenoid voltage cannot be measured without removing the connector. The sol terminal has a section that pushes in to release [push while pulling off the connector]

Have fun! But I'll guarentee you it is the starter. The top bolt is at the 11 o clock position on the transmission housing. You have to remove the battery to gain access to the top bolt. And if your lucky the bottom bolt will come out and won't strip [mine did]. I did it myself, but it took a few days.

From the bottom.

Remove passengers side spash guard. The sol terminal has a section that pushes in to release [push while pulling off the connector]

The top bolt is at the 11 o clock position on the transmission housing. You have to remove the battery to gain access to the top bolt. And if your lucky the bottom bolt will come out and won't strip [mine did]

Wow, excellent advice. Sounds like I'll have to plan a fair amount of time to do this. And I'll be careful with the bottom bolt - I stripped a belt tensioner bolt on my Buick and it was one of those projects where I invented a couple of new cusswords.
Thanks for the direction here.

The car wouldn't start at all now, not even keeping the foot on the brake while cranking. That trick worked great for a couple months although I still don't understand why. So last night I replaced the starter. I followed John Holl's advice and he was right about the method. I removed the battery, the battery cable bracket below the intake manifold, the passenger side wheel, the plastic engine shield just to get access to the starter. The top bolt was a bugger to find and is a blind location even using a shop mirror. If John hadn't mentioned its on the transmission housing, I might not have found it at all. That was a challenge to loosen and then re-tighten. But the operation was a success. The car started right up. The Chevy dealer wanted $399 for a new starter, and they would have had to order it. So I went to a local discount auto parts store and had them order a rebuilt one with a lifetime warranty for $160 plus a $35 core charge. It took a few days to arrive. Well worth it though. Thought I'd post the follow up to this and again thank John for his help.
Tom B.