If you sold your primary personal residence and you lived in and owned the home for at least two years in the five year period on the date of sale, you do not have to report the sale if your gains are less then the exclusion amounts of $250,000 if filing Single or $500,000 if filing Married Filing Jointly (and both lived in the home for two years). What you do with the proceeds of the sale is not relevant. Any interest earned in your saving account from the deposited funds has to be reported on your tax return.
The requirement to purchase another primary home after the sale of the old home to defer capital gains taxes was removed from the tax code in 1997.
If you had a gain greater then the exclusion amounts then you would have to report the sale. Also, if you received a Form 1099-S for the sale either with a gain or a loss, the sale has to be reported.
since your mother and father are part owners they're are responsible for reporting their pro-rata share of the sale and do not qualify for the exclusion if they did not live in the house as their principal residence for 2 out of 5 years before sale. if you get all the proceeds, they have made a gift to you of their pro-rata share of the proceeds and if greater than the annual gift tax exclusion amount they are supposed to file a gift tax return though most likely they will owe no gift taxes.
reporting as far as your 1040 taxes: if a 1099-S is issued with your SSN, then you need to report it on your 1040 because the iRS will note it wasn't reported and since it doesn't know whether any amount is taxable. you will likely receive a notice.
i should mention that is it's your SSN, you'll need to report gross proceeds per the 1099 then show a reduction for the pro rata share belonging to your parents. also cost will need to be prorated
the tax responsibility are the same regardless of when the cash was deposited. the other responders are answering that part of the question, but I think the point of your post is the fact that the cash ended up in your personal savings account temporarily. the taxes are based on who owned the property when it was sold and not where the cash was temporarily parked