Yes, parsing dates entered from external systems is a pain in Salesforce as the SFDC date parsing is limited. I use this Utility class method to parse dates in locale or yyyy-mm-dd format and it could be extended for other allowed formats

	//	--------------------------------------------------------
	//	parseDate; null is invalid Date; yyyy-mm-dd and locale-specific e.g. mm/dd/yyyy or dd/mm/yyyy formats supported
	//	--------------------------------------------------------
	public static Date parseDate(String inDate) {
		Date	dateRes		= null;
		//	1 - Try locale specific mm/dd/yyyy or dd/mm/yyyy	
		try {
			String candDate		= inDate.substring(0,Math.min(10,inDate.length()));// grab date portion only m[m]/d[d]/yyyy , ignore time
			dateRes 	= Date.parse(candDate);
		}
		catch (Exception e) {}

		if (dateRes == null) {
		//	2 - Try yyyy-mm-dd			
			try {
				String candDate		= inDate.substring(0,10);			// grab date portion only, ignore time, if any
				dateRes				= Date.valueOf(candDate);
			}
			catch (Exception e) {} 
		}
		
		return dateRes;
	}

	@isTest
	private static void testParseDate() {
		System.assertEquals(Date.newInstance(2020,1,1),		parseDate('2020-01-01'));
		System.assertEquals(Date.newInstance(2020,1,1),		parseDate('2020-01-01T01:09:00Z'));
		System.assertEquals(Date.newInstance(2020,1,1),		parseDate('01/01/2020'));
		System.assertEquals(Date.newInstance(2020,1,1),		parseDate('1/1/2020'));
		System.assertEquals(Date.newInstance(2020,1,1),		parseDate('01/01/2020 05:08:00.000-0800'));
		System.assertEquals(null,							parseDate(null));
		System.assertEquals(null,							parseDate(''));
		System.assertEquals(null,							parseDate('ab/de/1201'));
		System.assertEquals(null,							parseDate('13/01/2020'));
		System.assertEquals(null,							parseDate('2020-13-01'));
				
	}

Note that if you need to parse dates from a VF page, then I strongly recommend changing the VF apex:inputText component to apex:inputField and associate that inputField to an SObject field, even to the extent of creating new SObject fields of type Date if necessary.  In this way, you will always get a valid date from the user input and you won't care about these locale issues