Paycheck parking runs on a line that's secured by your house. So what happens when you sell the house? A question with a boring answer at closing and a more interesting one for your strategy.
The boring part first. At closing, escrow pays off every lien against the property from your sale proceeds, in order. First mortgage, then the HELOC, then you get what's left. You don't need to zero the line beforehand. The title company handles it, the lender releases the lien, and the line closes for good, because the account doesn't survive the collateral. Some lenders freeze new draws once they learn the home is listed, which is their right under the agreement, so don't count on drawing from the line during the sale window.
Now the interesting part: your hub just disappeared. If you park your paycheck and daily spending through that line, moving means running without the system for a stretch. Plan for three things. First, stop parking a few months before listing, and let the checking account carry the load the old-fashioned way. Second, don't chunk money into the line that you're about to need for the next down payment. Equity you send to the line comes back at closing anyway, so late-stage chunks mostly just tie up your flexibility. Third, expect a gap on the other side. A new HELOC on the new house needs seasoning, equity, and an approval process that looks at your fresh mortgage with fresh eyes.
The move itself is normal life, and the strategy should bend around your life, never the reverse. Treat the gap months as a scheduled pit stop. Keep the habits, the surplus tracking, and the monthly review running even with no line to review, because the household that keeps its numbers current restarts the machine in one month instead of six.