I will be paying for a private companion for my elderly father. Do I have to be concerned with paying employment taxes?

You generally must withhold Social security and Medicare taxes from wages you pay an employee, but not from money you pay an independent contractor. A worker is an employee if you can control not only the work they do but how they do it.

NO, they work for themselves. At the end of the you give them a 1099 form and they can claim it if they want to.
But, if YOU take money from your Dad for is care, you DO have to pay taxes on it. It rounds out to about 30% of your pay so save it separate to be safe. We have been doing this for a couple of years now with an accountant. Always pay in check for your records. We even have out woman make out and sign a time-sheet for our records.

This depends on how you are contracting the private companion. If you are paying an hourly wage as an employer and the weekly amount exceeds the amount exempted (check with a tax accountant or IRS for this) then you must pay the employer's share of Social Security and Medicare taxes. However, if you are contracting the companion on a temporary basis for a pre-agreed amount, then you can just provide a 1099-MISC form to the companion and he/she is responsible for any taxes owed on that income

I agree with John P Beyrer, and I have some more to add. The rules for defining scope of control must ALSO CONFORM to the labor laws of YOUR STATE! Many people get in trouble with the independent contractor thing. Most of these situations DO NOT conform to independent contractor definition. If you, the employer, control when, where, and how the work is performed, YOU HAVE A W-2 ELIGIBLE EMPLOYEE. Period.
A lot of lawmakers and wealthy people are in BIG TROUBLE with the IRS, MEDICARE, and SSA, and get handed a BIG TAX BILL with penalties, interest, and sometimes, FINES. Then their STATE goes to work on them for unpaid employer unemployment taxes, failure to withhold state income taxes, and failure to establish and contribute to a workers compensation fund or show bond/self insured status.

Please READ IRS Publication 926 carefully. I think when you are done, you might want to consider hiring through a service, like Visiting Angels, etc. or hiring a accountant to handle the details of required fiduciary payments of withheld taxes and employer contributions. Making those payments late is a huge and expensive misstep.

I have seen these situations personally, and seen people try to get away with payment with in kind housing etc...and get slammed HARD, in tax audits.
https://www.irs.gov/publications/p926/ar02.html

If we hire several caregivers and pay them by cash or check, do we need a contract with them that as PRIVATE CONTRACTORS?

Again, if you Control when, where, how in the employment issue, you cannot consider them 1099 independent contractors. They are domestic employees.

the goverment will tsx us for any thing ,chck with the irs office

If you give them a 1099 MISC form do you show that you gave it somewhere on your tax form?