The 7-sphere is the highest dimensional sphere that is parallelisable, and the next highest is the 3-sphere. An inherent property of each of these spheres is that they're embedded in a space of dimensionality at least one greater than their own: 4 and 8.
We know that quaternions and octonions are important algebras in mathematics in general and physics which have properties not found in other dimensions.
Is this property of parallelisability derived from the spheres themselves, or from the 4- and 8- dimensional spaces in which they're embedded?
If it's derived from the spaces, are there other special properties these spaces share as a result and are there other examples of mathematical objects: shapes, knots, networks, algebras, combinatorics, vectors, polynomials etc. which are parallelisable or have some other analogous property only in dimensions 4 and 8?
An $n$-dimensional manifold is parallelisable if and only if there exists $n$ vectors fields $X_1,...,X_n$ defined on $M$ such that for every $x\in M$, $X_1(x),...,X_n(x)$ is a base of $T_xM$, so to be parallelisable by definition does not depend of the manifold where $M$ can be embedded. Remark that if $M$ is compact, it can be embedded in $R^{2n+1}$ by using Withney theorem.
Concerning the sphere $S^3$ and $S^7$, the structure of $R^4$ and $R^8$ can be used to show that they are parallelisable. You can define $S^3$ as the set of quaternions of module $1$, the multiplication of the set of quaternion induces on $S^3$ a structure of a Lie group, this shows that $S^3$ is parallelisable.