Understood the requirements, but wouldn't mind having a look at the sample sites you like to get an even better idea.
Relevant Skills and Experience
I've built many dynamic web apps, so I am confident you'd be happy with the outcome. I would code it in PHP/MySQL (Codeigniter framework)
Proposed Milestones
$150 AUD - initial
$600 AUD - parent/carer registration and login, user profile settings, parents can search/filter the carers
$600 AUD - contact carer, parent post job, carers can filter jobs posted by parents
$300 AUD - carers can apply for jobs, parents can select a carer to work with - starting the project
$300 AUD - carer can generate an invoice for parent, parent can pay online, carer receives the payment notice
$494 AUD - on completion
The design would be nice, clean and responsive using bootstrap4. Code would be clean and easy to maintain and expand. I am happy to do any upgrades or maintenance outside of the initial project scope for $25/h, or as a new project for a fixed cost.
In case you want to work with me, it would be great if we clearly defined the scope of the project upfront. You already did that for the most part, but there are probably more details which would be useful to be said before starting.
I would share the testing URL on my server so you can observe the progress of the project, and hopefully provide feedback in order for us to stay on the right track.
Regarding online payments, I assumed you want paypal, so I quoted for that only, I can implement other means of online payments too.
For SMS, I believe we'll use an existing API for that, I have worked with a few and as long as they have a well documented API I don't mind which service we'll use.
Regards,
Julijan