Solve washer/dryer closet outlet issue. Lacks 110v outlet, need to know my options
What to watch for in replacing washer/dryers in developments

The typical home washer-dryer set needs SEVEN utilities plumbed to it:   120VAC, 240/120VAC (both dedicated circuits), cold water, hot water, dryer vent, sanitary sewer, and gas line.   Needless to say, in large multi-unit installations, the builders commit some brain cells to reducing that number to three: 240V power, cold water and sewer.   How do they do this? 


They eliminate gas by requiring electric dryers. 
They eliminate dryer vent by using condensing dryers. 
They eliminate hot water by using washing machines that heat their own water.  
They eliminate 120V one of two ways: 


They use a single-unit "combo" appliance that figures out internally how not to overload the circuit. 
They use separates (because buyers like the sight of separates), but the dryer is extremely special, and provides a 120V outlet for the washer, and somehow has internal controls e.g. that draw less current from the washer's leg, to assure no breaker trips. 



You have to be an educated buyer

Here's where such apartment dwellers get into trouble: They prance over to the appliance store and decide "I can get anything I want!" and just go "One of those, and one of those, whee!"

And of course they get them home and discover no hot water, no dryer vent and/or no 120V.  Whoopsadaisy! 

So you will need to start this enterprise by either consulting with management or doing a careful survey of exactly which utilities you do have.  And then you will need to make a decision as to whether to ante up the hi-3/low-4 digit cost of getting those utilities plumbed in (dryer vent, expect a whole lot more; gas ain't gonna happen) ... or to go back to the appliance store with a list of the utilities  you do have, and narrow the list of options accordingly. 

I suspect there'll be a few local appliance stores who are very familiar with your particular development and know exactly what you need. 

Hork-a-dork adapters aren't going to cut it

Right off the bat, they'll trip your breaker when you try to run both appliances at once.  Further, the choice of adapters is a lot more limited than you realize, as I'll discuss below.  And lastly, it's good money thrown after bad - you're spending fine hard-earned coin so you can force a bad choice of cheaper unit into a place it simply doesn't belong.  You could be $300 down the "cheap splitter" rabbit hole before you find stuff that actually works for you, and even then, your performance will be degraded.  You'd have been wiser to put the $300 toward provisioning a proper circuit or buying the correct appliances.  

There's also a huge supply problem with those cheaters.  

And by the way, the dual 240V splitter is not necessary; it is already legal to fit 2 proper NEMA 14-30 receps on the same circuit permanently installed in or on the wall.  An electrician can do that easily.  Might not like the idea too much, but is legally able. 

Just say no

Finally, don't let anything touch mains AC wires that doesn't have a UL listing.  That's a UL in circle.  You may also substitute any of the NRTLs: CSA and ETL are the ones you're likely to encounter.  Here is what is not an NRTL: CE (China Export really), CCC (China's attempt to locally sanction a fake mark), RoHS (lead-free solder etc.), and FCC (won't jam out radio stations). 


Note that your Parkworld cable does NOT have a UL listing.  They say a lot of lies about how they use UL parts (not good enough) and built it to UL spec (where's the stamp then?) but no. The fact that they would deceive about this is even more horrifying. This is why they're selling on Amazon's junk shop, even Home Depot wouldn't let this in the store! 
Generally buying mains electrical parts on Amazon is a terrible idea. Avoid!  Even if you sidestep the Amazon Marketplace flea market, you're running the risk of getting counterfeits due to "Commingling". 


The Home Depot splitter you found is less alarming because Home Depot generally insists on UL-approved items - they're a retailer not a "platform", so they do have liability for what they sell.