Short answer: Yes
Long answer: Maybe
Longer answer Sometimes
Most accurate answer: It depends
Starting with requirements, the question would be, do you need to pursue an agile development cycle? In agile, you will be looking to produce a minimum marketable unit with each sprint. With classic waterfall, you're more likely to want to have everything spec'ed out up front.
Either way, with solid requirements, many things can be done in parallel. Designers can start mocking up views. Front-end developers can start stubbing out pages, navigation, security, etc. Back-end developers can start building data structures and stubbing out services.
With fewer staff, you may have to pick where you start based on the needs of the project. Personally, when I'm doing full-stack development I tend to pick vertical slices of the application and take multiple passes at layering in functionality. Generally I start at the bottom since you can't really get the views working properly until the foundation is in place. Even then, you can start with empty services so you can at least know the pieces are talking to each other. That can all change depending on reasons.