Asking for different leave schemes during probation?

I'm interviewing for a public services job with good pay but it's a very awkward time for me childcare wise, I have 1 year left before my youngest can access full afterschool (he's in ecce at the minute). I do have a childminder that collects him and all but ecce is pretty strict if kid is sick they're off for 48 hours so I'm worried about this year.

I think having something like parental leave where you take 1 day a week on a Friday would give me a lot of flexibility, there are so many half days and school closures on Fridays as well. I know plenty of people in the public sector that space parental leave like this but I know it's not meant to and it's up to the discretion of the employer.

Would it be awkward if I got the job to ask for that out of the gate? The probation period is likely going to be a year. I don't think working from home or possibly even flexitime will be available until after the first year either.