We will never know the exact numbers. However, from reported figures we know that $SBUX, $DNKN, $PNRA, $MCD all have similar margins across gross, ops and P&L (50-70, 10-20, 10-20 respectively).
the goal of all fast food centers is to produce a unit cost as close to $1, preferably lower, as possible and we also know from reported figures that 1 cent is the expected associated labor cost of a starbucks unit.
Knowing that the price of milk on commodities market is 16.42hwt or 1 cent / oz, knowing that SBUX coffee beans are 7cent/oz we can extrapolate that suitable extra costs for alternative milks must be in the single figure cent range.
Further supported by how if you are to go to a post-supply-chain-shipping-and-procurement wholesale vendor then the price of oatly barista edition oatmilk is 10c/oz and we can very safely assume that SBUX gets it much MUCH cheaper so we at least know the ceiling is $0.1
So, while I was exaggerating for effect in my original reply, the actual numbers- even if they are paying the same price as I would walking into a wholesaler (EXTREMELY unlikely):
price of 16oz cow milk latte: $4.25, unit cost $1, milk cost 1c
So in the extreme worst case scenario for starbucks they are making an extra 6% profit per ounce on oatmilk over cow milk, so not at cost-to-price parity.
And that's the worst case, they are probably making more.