I haven't tried asking but I think it would be a big no.
Please take my child(ren) to work day was started (hopefully in semi-seriousness) to draw some attention to the plight of SAH-parents everywhere who don't have the perk of a day/night/minute off in 365 days. OK, that might be a bit of an exaggeration but most of the time 24.7.365 sums up life as a mum at home. Wee Guy is in school now (well, he was last week before the summer 'holidays' started - note how holiday is in quotation marks) and yes, he has a day of daycare so I technically get a few hours 'leisure time' (again note the quotation marks) but if he's sick, off colour or just plain being non-sociable that day it isn't daddy who fills in. And it's daddy who earns the salary so this is The Wisest Choice.
But I have noticed a distinct lack of credible time off, career advancement, regular breaks scheduled in my latest employment choice hence making the most of alternate or surrogate parenting arrangements which get me some head space.
Take Your Kids To Work Day is something that all at-home parents do every day of the year (unless I somehow lost the memo reminding me that being a stay-at-home parent is not really work). So maybe today should be renamed Please Don't Leave My Kids At Home With Me Today day or possibly even Please Stay At Home With My Kids While I Get To Leave The House In Less Than 15 Minutes And Whoop It Up Without Them day?
There must be a Hallmark card for that somewhere ...